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FINDING THE nth TERM IN A SEQUENCE

Updated on October 4, 2010

Formula's for the nth term

ALGEBRA

Here I intend to look at what is known as the nth term sequence. In doing so we will look at some examples and find the nth term which is known in mathematical terms by a "u" followed by a subscript(small) "n".

So what exactly is the connection( the pattern) between the "n" and the "un".

If we take a closer look at the un we can see that it goes up in two's (2's).

(Position) U         1    2    3    4    5

 ( Term)    Un          3    5    7    9   11

      n x 2               2    4    6    8   10

 

3 + 2 =   5

5 + 2 =   7

7 + 2 =   9

9 + 2 = 11

 

Therefore we can write down that un = 2n, in other words we are doubling the number.

Looking again we may need to adjust that 2n slightly;

2n means 2 x n therefore,

2 x 1 =   2

2 x 2 =   4

2 x 3 =   6

2 x 4 =   8

2 x 5 = 10

These are the numbers we have got after multiplyingn x 2, you will have noticed that they are 1 less in each case than the numbers in the unbracket, therefore we need to add on 1 in the formula.

Our formula now looks like this un = 2n + 1 (where the un is the nth term)

2 x n =   ? + 1 =  un

2 x 1 =   2 + 1 =   3

2 x 2 =   4 + 1 =   5

2 x 3 =   6 + 1 =   7

2 x 4 =   8 + 1 =   9

2 x 5 = 10 + 1 = 11

Answer:- un = 2n + 1

 

This is the type of question for the nth term, could be asked in an exam paper;

Find the nth term for the sequence; 2, 6, 10, 14, 18 ... (etc)?

To begin with we need to change this sequence into a table!

The "n" is the position and the “un” is the term.

The position simply represents the number of digits in a sequence, in the above there are "5" [ 2(which is 1), 6(which is 2), 10(which is 3), 14(which is 4), 18( finally this is number 5 )]

(Position) U          1    2     3     4     5

  ( Term)   Un            2    6   10   14   18

     n x 4                 4    8   12   16   20

So what exactly is the connection (the pattern ) between the "n" and the "un", in this question?

If we take a closer look at the "un" we can see that it is going up in four's (4's).

  2 + 4 =   6

  6 + 4 = 10

10 + 4 = 14

14 + 4 = 18

Therefore we can write down that un = 4n (which means 4 x n)

4 x 1 =   4

4 x 2 =   8

4 x 3 = 12

4 x 4 = 16

4 x 5 = 20

These are the numbers we have got after multiplying n x 4, you will have noticed that there are 2 more in each case than the numbers in the un term, therefore we need to subtract 2 in the formula.

Our formula now looks like this; un = 4n - 2 (where the unis the nth term)

4 x n =   ? - 2 = un

4 x 1 =   4 – 2 =  2

4 x 2 =   8 – 2 =  6

4 x 3 = 12– 2 = 10

4 x 4 = 16– 2 = 14

4 x 5 = 20– 2 = 18

Answer:- un = 4n - 2

 

In this example we need to find the nth term for this sequence;

Find the nth term for the sequence; 3, 6, 11, 18, 27 ... (etc)?

To begin with we need to change this sequence into a table!

This example is quite similar to the one above!!!!

LOOK CAREFULLY:-

The "n" is the position and the “un“ is the term

Again there are five numbers and they are holding the first five POSITIONS

The position simply represents the number of digits in a sequence, in the above there are "5" [ 3(which is 1), 6(which is 2), 11(which is 3), 18(which is 4), 27( finally this is number 5 )]

(Position) U           1    2     3     4    5

  ( Term)   Un             3    6   11   18   27

n2=(n x n)              1    4     9   16   25

So what exactly is the connection (the pattern ) between the "n" and the "un", in this question?

Is there a set pattern?

Are they going up in 2's or 4's? --- NO they are not!

  3 + 3 =   6

  6 + 5 = 11

11 + 7 = 18

18 + 9 = 27

In a case like this we would square (n2) the numbers and see what that brings us too.

Therefore we can write down that un = n2 (which means n x n)

1 x 1 =   1

2 x 2 =   4

3 x 3 =   9

4 x 4 = 16

5 x 5 = 25

These are the numbers we have got after multiplying n x n, you will have noticed that they are 2 less in each case than the numbers in the un term, therefore we need to add 2 in the formula.

Our formula now looks like this; un = n2 + 2 ( where the un is the nth term)

n x n =  ? + 2 = un

1 x 1 =   1 + 2 =   3

2 x 2 =   4 + 2 =   6

3 x 3 =   9 + 2 = 11

4 x 4 = 16 + 2 = 18

5 x 5 = 25 + 2 = 27

Answer:- un = n2 + 2

 

Another favourite question in exams is as follows;

The nth term of a sequence is represented by this formula:- un = 3n + 2

We are asked to;

(i) Find the first 4 terms

(ii) To find the 49th term

( i ) Find the first 4 terms

Our formula is:- 3n + 2 = un

Therefore our 1st term is u1 = 3 x 1 = 3 + 2 =   5

                             2nd term u2 = 3 x 2 = 6 + 2 =   8

                             3rd term u3 = 3 x 3 = 9 + 2 =  11

                            4th term u4 = 3 x 4 = 12 + 2 = 14

Our first four terms are; u1 , u2 , u3 , and u4

They end up as 5, 8, 11, and 14

 

\ First 4 terms are: 5, 8, 11, 14. (Answer:-)

 

 

( ii ) We have been asked to find the 49th term.

49th term: This will be known as u49

As before in part ( i ) the formula is; un = 3n + 2

Therefore we will replace the “n” with the 49

 u=       3n             + 2

u49 = (3 x 49) = 147 + 2 = 149

(Answer) u49 = 149

Comments

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    • profile image

      artina 

      5 months ago

      what is the nth term of 12,23,36,51,68

    • profile image

      john 

      3 years ago

      find the nth term of 2n+5. find the frist term,second term and common difference

    • profile image

      amberb 

      3 years ago

      HELP ASAP the first 5 terms and the 100th term for the sequence 0.5n-0.5

    • profile image

      thalene 

      4 years ago

      the sum of the 17 term of an arithmetic sequence is 2414 and the last term is 256. find the first term

    • profile image

      wow 

      4 years ago

      Not bad

    • profile image

      pooja 

      4 years ago

      This website is boaring

    • profile image

      Avicii 

      4 years ago

      Does anybody on this goddamn blog answer questions!!!??

    • profile image

      Telamon 

      4 years ago

      what is the first 5 terms and the 100th term for the sequence whose rule is 0.5n - 0.5

    • profile image

      Rues 

      4 years ago

      24,54,103,....

    • profile image

      becky 

      5 years ago

      how would I solve for the nth term:

      1+2=(2x3)/2

      1+2+3=(3X4)/2

      1+2+3+4=(4x5)/2

    • profile image

      Naseem Choudhury 

      5 years ago

      what is the term of 25-4n

    • profile image

      kola 

      5 years ago

      1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20=210

    • profile image

      tom 

      5 years ago

      what's the nth term for 10, 40, 90, 160, 250, 360, 490...

    • profile image

      ray 

      5 years ago

      what's 5 ways to find the 77th term?

    • profile image

      Cherry 

      6 years ago

      N term of these number 13,21,37,69,133. Please help

    • profile image

      Kiki 

      6 years ago

      Given the progression 7, 12, 17 and 22. What is the expression for the nth term of the progression?

    • profile image

      Howard 

      6 years ago

      what is the pattern of these numbers?

      a)2,3,6,85,68

      thank you

    • profile image

      timmi2000 

      6 years ago

      wondering if anyone could help me with these questions, and help me work it out as i do not seem to get it.

      1) find the 8th term of each of these sequences

      a) 2,5,8,11

      b) 3,7,11,15

      c) 2,10,50,250

      d) 2,6,18,54

      2) find the sum of the first 10 terms of the following series

      a)3,8,13,18

      b)1,3,9,27

      3) the first term of an arithmetic series is 1. the common difference is 6 find the 10th term of the series, the sum of the first N terms is 7400 find N

    • profile image

      met 

      6 years ago

      what are the next two numbers 4, 5, 14, 15, 40, 41, _, _, ?

    • profile image

      luiza 

      6 years ago

      Please help me

      look at the equation n+3=12. Use iot to work out the value of n-3

    • profile image

      Kyle 

      6 years ago

      Which statement describes how to find the nth term in the sequence?

      12, 29, 46, 63, 80, ...

      A.add 5 times the term number and 12

      B.subtract 17 times the term number from 12

      C.subtract 5 from 17 times the term number

      D.add 5 and 17 times the term number

    • profile image

      Adam Saleh 

      6 years ago

      The nth term of a sequence is n^2-6n-4, find the sum of 3rd and 4th terms

    • profile image

      chris 

      6 years ago

      An arithmetic sequence has first term a=6 and common difference d=12. How many terms of this sequence must be added to get 5040?

    • profile image

      Pavan Kumar 

      6 years ago

      24 24 28 46 94 what is the next sequence

    • profile image

      ragu 

      6 years ago

      In the sequence, 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 when we add any three numbers we have to get the answer 30

    • profile image

      yaya 

      6 years ago

      hi am year 7 what is the 5 terms and the 10th term of 2n+2

    • profile image

      fawzy 

      6 years ago

      plz define in detail what do we do with sequences not having a common difference E.G 0..-1..-8..-27..-64.

      And also do you happen to know any formula like

      Tn= A+d(n-1)+ 1/2[(n-1)(n-2)c] which claims to work on everything. If Yes than please explain the formula.

    • profile image

      Helmi 

      6 years ago

      how to solve this

      U 1 2 3 4 5

      Un 1 4 10 22 46

    • profile image

      Lisa 

      6 years ago

      trying to find out the correct sequence for if n+1/3

    • profile image

      Subhasish Raha 

      6 years ago

      what is the missing number in the following series

      9,19,40,___,146?

    • profile image

      suruthi 

      6 years ago

      if the sum of 4 consecutive terms in a gp is 60 and if the t1,18,t4 are in ap find the terms

      ans(4,8,16,18)

    • profile image

      Kalyanakrishnan 

      6 years ago

      give solution to this sum

      sum of the series 2+4(3*3)+6(5*5)+.........+22(21*21)

    • profile image

      Jimmy 

      6 years ago

      nth term of sequence the sequence is 4, 12, 24

      This is what i gett

      4 12 24 40

      \ / \ / \ /

      8 12 16

      \ / \ /

      4 4

      But I dont know how to make a formula from this help

    • profile image

      SURUTHI 

      6 years ago

      HI,

      pleasssssssse solve this problem ,i am not getting the SOLUTION anywhere in the net.

      "the sum of the first n terms of an ap are in the ratio (3n +8):(7n + 15), find the ratio of the 3 rd terms"

      {ans=23:50}.

    • profile image

      SURUTHI 

      6 years ago

      HI,

      pleasssssssse solve this problem ,i am not getting the solution anywhere in the net.

      "the sum of the first n terms of an ap are in the ratio (3n +8):(7n + 15), find the ratio of the 3 rd terms" ans 23:50.

    • profile image

      CHRISTY 

      6 years ago

      nth term of sequence ?7, ?21, 3?7.....

    • profile image

      williamovely@yahoo.com 

      6 years ago

      a series is s

      uch that the sum of the first nth therms is 2n^2 minus 3n.

      1.how to Prove that the series is arithmatic

      2.find the value of the 18th term

    • profile image

      Fazla 

      6 years ago

      Maa''m i can't understand general term.. I don't knw why.. If u can help me in dis.. it will be a great help..

      -2,4,-8,16.. can u help me 2 find da general term of dis set??

    • profile image

      dmndmz 

      6 years ago

      Greetings ;)

      What is the formula for the sequence 16, 29, 48, 52, 54 and 5 ?

    • profile image

      DIANA 

      6 years ago

      WHAT IS THE FORMULER FOR THE SEQUENCE; 22,18,14,10?

    • profile image

      paul 

      6 years ago

      in an AP the fifth term is 8 and the nineth term is 14. fin the first term.

    • profile image

      brittany 

      6 years ago

      Can Anyone Find What's The 6th Term For 5 , 7 , 11 ....

    • profile image

      john 

      6 years ago

      terms: 1,2,3,4,5,...n

      values: 1,2,4,8,16,...

      please find me the formula?

    • profile image

      waleedrehman 

      6 years ago

      plz find the indicated term of the following,1,3/2,5/4,7/8,...a7.

    • profile image

      alicia 

      6 years ago

      what is the 75th term 8, 14, 20, 26, 32

    • profile image

      Jim 

      6 years ago

      The nth term of a sequence is the l.c.m of integers from 1 to n. Which term of the sequence is the first one that's divisible by 100?

    • profile image

      Charlott3_Fay3 

      6 years ago

      What is 2-3G and 7-1/2n??? I'm really confused and it's annoying me please answer tonight xxxx

    • profile image

      anonymous 

      6 years ago

      given the general form an^2+bn+c

      find a

      pls help with the above

    • profile image

      6 years ago

      What is the Nth Term to: -1, 7, 18, 29, 42, 57

      and 2, 9, 18, 29, 42, 57?

      Please help!

    • profile image

      6 years ago

      5 x 4 x 3 ... -5 x -2 x - 4 Write out the correct form of the sequence

    • profile image

      sita 

      6 years ago

      how to find the 40 term of a sequance with term to term rule add 5and the 1st term of 6

    • Anne David profile image

      Anne David 

      6 years ago

      and the tenth term of 0.4,0.04,0.004,,, plz give me a formula for this

    • Anne David profile image

      Anne David 

      6 years ago

      what is the formula of sixth term of 3/4,3,12...

    • profile image

      t.elia 

      6 years ago

      how bout sixth term of 3/4,3,12...

    • profile image

      Geralyn. 

      6 years ago

      the radiator of the truck contains 12 gallons of water. We draw off 3 gallons and replace it with alcohol, then we draw off 3 gallons of the mixture then replaced it by alcohol. until 8 drawings and replacements have been dine. How much alcohol is in the final mixture?

    • profile image

      Ronak 

      6 years ago

      what will be the next term to the series 2,3,4,9,16... and how it will be???.....

    • profile image

      shubham 

      6 years ago

      iam confuse between nth and n terms of A.P

      PLEASE HELP T.ELIA

    • profile image

      akanksha raj 

      6 years ago

      what will be the next term for this series 2,3,4,7,13,12,12,23,20,17,33,28,? please help.

    • profile image

      Vin 

      6 years ago

      What's The Next Three Terms of the Arithmetic Sequence?

      6,12,18,24,__,___,___?

      Help Please :D

    • profile image

      Filip 

      6 years ago

      -1 , -2 , -1 , 2 , 7 , 14 what's the nth term please?

    • profile image

      Simon 

      6 years ago

      I'm stuck on this question so could you help me out?

      All terms in a geometric sequence are positive integers, and the first three ter4ms are n, n+3, and 2n+6. What is the fourth term of this sequence?

    • profile image

      arihant 

      6 years ago

      what is the nth term of 1+(1+2)+(1+2+3).....to n terms

    • profile image

      joyce 

      6 years ago

      what would be the nth term of 30,28,26,24,22..

    • profile image

      joyce 

      6 years ago

      what would be the nth term of 30,28,26,24,22... and what would be the nth term of 40,37,34,31,28...

    • profile image

      grace 

      6 years ago

      what would be the nth term of 40,38,36,34,32...

    • profile image

      rohit choudahry 

      6 years ago

      0,2,4,6,8,12,12,20,16,?

      answer is 12.

      can u explain the trick behind this?

    • profile image

      eivlys 

      6 years ago

      Find the next term of the progression 2 4 6 14 23

    • profile image

      Carl 

      6 years ago

      Hi would you be able to help me with the general term (tn) I guess, I doesn't know the word in english... well here are the numbers, 2,7,15,26,40 thanks :)

    • profile image

      Kieran 

      6 years ago

      What is the nth term for this sequence (or similar ones): 2, 7, 15, 26. It goes up like this: +5, +8, +9.

    • profile image

      miley 

      6 years ago

      how do you find the equation and 50th term of 1,4,9,16,25,36,48

    • profile image

      Marc 

      6 years ago

      Find the next term of the progression 24,12,8,.....

    • t.elia profile imageAUTHOR

      t.elia 

      6 years ago from Northern Ireland

      Simba,

      What are you working on multiplication?? Please clarify.

      Telia

    • t.elia profile imageAUTHOR

      t.elia 

      6 years ago from Northern Ireland

      MAHI,

      Arithematic Progression:-

      A sequence of numbers. Let's start with a number: a1.

      Now add a number d, (for "difference").

      We get a1 + d and the first 2 terms in our sequence are:

      a1, a1 + d

      For the next term, let's add d to that last term and we have a1 + 2d.

      Our sequence is now:

      a1, a1 + d, a1 + 2d

      We continue this process and the resulting set of numbers is called an arithmetic progression (AP) or arithmetic sequence

      Good Luck Telia

      Sorry posted the comment and wasn't signed in.

    • profile image

      MAHI 

      6 years ago

      WHAT IS AP?

    • profile image

      ogleoo 

      6 years ago

      i keep forgetting that n means term number!!!

    • profile image

      ogleoo 

      6 years ago

      thank you so much

    • t.elia profile imageAUTHOR

      t.elia 

      6 years ago from Northern Ireland

      Ogleoo,

      48 - 4n

      48 - (4 x 1)= 48 - 4 = 44

      48 - (4 x 2)= 48 - 8 = 40

      48 - (4 x 3)= 48 - 12 = 36 etc

      I hope you find this easy to follow

      Good Luck

    • profile image

      student 

      6 years ago

      hello plaese help me find Recurrence Relations

      t(n)=t(3n/5)+t(2n/5)+n

    • t.elia profile imageAUTHOR

      t.elia 

      6 years ago from Northern Ireland

      Hi Paul, this link you provided isn't working.

    • profile image

      Paul 

      6 years ago

      My homework sheet says that i have to write a rule in terms of n, i don't know what that means, the website for my homework is http://docs.google.com/viewer?a=v&q=cache:ap2t.../maths/docs/building_number_patterns.doc+building+investigative+skills&hl=en&gl=uk&pid=bl&srcid=ADGEESgdc2qKUJbnfZK9K6L22BqBGd27GIaVEhDj6m_QE5xBedMtgqj5QD7hfd1vx0armiQYpFn4mPs83vidpckpAmTdbMgP_2PR7u30s46xICSkUbnCcSq1TaTrXxr-ywh0FKdy2oaH&sig=AHIEtbScDxgftte_31i6WwekjrGnqCDRgA, i also don't get question 4

    • t.elia profile imageAUTHOR

      t.elia 

      6 years ago from Northern Ireland

      Hi Jamie

      Your answer is 3n + 9

    • profile image

      Jamie 

      6 years ago

      Hi, can you help me.

      If i have these 4 numbers, 12 15 18 21, what's the sequente to get the nth term?

    • profile image

      shreyansh 

      6 years ago

      The value of (2/3)(3/4)(4/5).....(1-1/n)=

    • t.elia profile imageAUTHOR

      t.elia 

      6 years ago from Northern Ireland

      Oscar,

      Answer is 3n + 4

      3 x 1 = 3 + 4 = 7

      3 x 2 = 6 + 4 = 10

      3 x 3 = 9 + 4 = 13 etc

      Good Luck & hope this helps

    • profile image

      oscar 

      6 years ago

      can you solv please 7,10,13,16

    • profile image

      bubble 543 

      6 years ago

      what would the answer be for the nth term for a sequence is n-2/ n squared

      write the first three terms of the sequence

    • t.elia profile imageAUTHOR

      t.elia 

      6 years ago from Northern Ireland

      LIV

      26-4n

      26 - 4 x N

      26 - 4 x (1) = 22

      26 - 4 x (2) = 18

      26 - 4 x (3) = 14 etc

      Hope this helps

      Good Luck

    • profile image

      liv 

      6 years ago

      26-4n?

    • profile image

      Mahmoud Shahoud 

      6 years ago

      Hi , You're smart , but all comments don't help me .

      can you help me .

      I have this sequence .

      1 1 1 1 1 1 1 1 1 2

      I want to calculate the 11st element and 12nd element

      I want Math Equations .

      I have a product gave me the answer but I want the way .

      11st term = 11

      12nd term = 56

      please help me .

    • t.elia profile imageAUTHOR

      t.elia 

      6 years ago from Northern Ireland

      DAVID

      53, 56, 61, 68

      first difference is 3, 5, 7

      second difference is 2

      Tn = a + (n - 1) d

      The Sum of the n terms is written as Sn and we know that the nth term = Tn = Sn - (Sn-1)

      Tn = 53 + 3 + 5 + 7

      Tn = 53 + n/2[ 2*3 + (n - 1) 2]

      Tn = 53 + n/2 x 6 + 2(n - 1) (Cancel down)

      Tn = 53 + 3n + n^2 - n

      Tn = n^2 + 2n + 53 (Answer)

      Good Luck

    • t.elia profile imageAUTHOR

      t.elia 

      6 years ago from Northern Ireland

      DAVID

      50, 48, 46, 44

      Tn = a + (n - 1) d

      a = 50

      d = 48

      48 - 50 = -2

      Tn = a + (n - 1) d

      Tn = 50 + (n - 1) -2

      Tn = 50 - 2n + 2

      Tn = 52 - 2n (Answer)

      Good Luck

    • t.elia profile imageAUTHOR

      t.elia 

      6 years ago from Northern Ireland

      DOM & DAVID,

      you have both asked for the same problem here;

      3, 8, 15, 24, 35

      First difference is 5, 7, 9, 11

      Second difference is 2

      The Sum of the n terms is written as Sn and we know that the nth term = Tn = Sn - (Sn-1)

      Tn = a + (n - 1) d

      Tn = n/2 [ 2a + (n - 1) d ]

      Tn = n/2 [ 2*3 + (n - 1) 2]

      Tn = n/2 x 6 + n/2 x (n - 1)2 (cancel down)

      Tn = 3n + n^2 - n

      Tn = n^2 + 2n (Answer)

      Good Luck

    • profile image

      David 

      6 years ago

      Excuse me, would you help me with 50,48,46,44 and 3,8,15,24,35 and 53,56,61,68 please, thanks.

    • profile image

      dom 

      6 years ago

      3,8,15,24,35 what is the rule for that? im stuck !

    • t.elia profile imageAUTHOR

      t.elia 

      6 years ago from Northern Ireland

      RUSS & ME,

      Here is help with the sequence for your formula;

      1 , 8 , 9 , 12

      1^2, 2^3, 3^2 , (2*3*2) = 12

      So the first number is squared = 1

      ...the second number is cubed = 8

      ...the third number is squared = 9

      ...the fourth is the powers multiplied (2*3*2) = 12

      The fifth number will be 5^2 = 25

      The sixth number will be 6^3 = 216

      The seventh number is 7^2 = 49

      The eighth will be the powers multiplied (2*3*2) = 12

      and so on ......

      Hope this helps

      Good Luck

    • profile image

      Me 

      6 years ago

      WHAT IS THE FORMULA FOR tHIS SEQUENCE PLEAAAASE! 1 8 9 12

    • profile image

      Russ 

      6 years ago

      Can somebody help me with the sequence of 1 8 9 12 please...are there 2 sequences? What follows after 12? Much appreciated!

    • profile image

      Bex 

      6 years ago

      What's the nth term for: -1 +1 -1 +1 -1.

      Doing my GCSEs and stuck on homework.

    • profile image

      renee 

      6 years ago

      what are the first 5 nth term of 3n+6

    • profile image

      saf 

      6 years ago

      ok i know this is random but if you start with 2 and double each time then what is the number if the nth term is n+3

      then 105-5n

      2n-0.5v

      thanx

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