# 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

(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

16

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## Popular

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• artina

2 years ago

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

• john

4 years ago

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

• amberb

5 years ago

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

• thalene

6 years ago

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

• wow

6 years ago

• pooja

6 years ago

This website is boaring

• Avicii

6 years ago

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

• Telamon

6 years ago

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

• Rues

6 years ago

24,54,103,....

• becky

6 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

• Naseem Choudhury

7 years ago

what is the term of 25-4n

• kola

7 years ago

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

• tom

7 years ago

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

• ray

7 years ago

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

• Cherry

7 years ago

• Kiki

7 years ago

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

• Howard

7 years ago

what is the pattern of these numbers?

a)2,3,6,85,68

thank you

• timmi2000

7 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

• met

7 years ago

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

• luiza

7 years ago

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

• Kyle

7 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

7 years ago

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

• chris

7 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?

• Pavan Kumar

7 years ago

24 24 28 46 94 what is the next sequence

• ragu

7 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

• yaya

7 years ago

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

• fawzy

7 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.

• Helmi

8 years ago

how to solve this

U 1 2 3 4 5

Un 1 4 10 22 46

• Lisa

8 years ago

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

• Subhasish Raha

8 years ago

what is the missing number in the following series

9,19,40,___,146?

• suruthi

8 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)

• Kalyanakrishnan

8 years ago

give solution to this sum

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

• Jimmy

8 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

• SURUTHI

8 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}.

• SURUTHI

8 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.

• CHRISTY

8 years ago

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

• williamovely@yahoo.com

8 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

• Fazla

8 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??

• dmndmz

8 years ago

Greetings ;)

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

• DIANA

8 years ago

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

• paul

8 years ago

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

• brittany

8 years ago

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

• john

8 years ago

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

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

• waleedrehman

8 years ago

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

• alicia

8 years ago

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

• Jim

8 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?

• Charlott3_Fay3

8 years ago

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

• anonymous

8 years ago

given the general form an^2+bn+c

find a

pls help with the above

• 8 years ago

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

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

• 8 years ago

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

• sita

8 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

8 years ago

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

• Anne David

8 years ago

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

• t.elia

8 years ago

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

• Geralyn.

8 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?

• Ronak

8 years ago

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

• shubham

8 years ago

iam confuse between nth and n terms of A.P

• akanksha raj

8 years ago

• Vin

8 years ago

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

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

• Filip

8 years ago

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

• Simon

8 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?

• arihant

8 years ago

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

• joyce

8 years ago

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

• joyce

8 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...

• grace

8 years ago

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

• rohit choudahry

8 years ago

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

can u explain the trick behind this?

• eivlys

8 years ago

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

• Carl

8 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 :)

• Kieran

8 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.

• miley

8 years ago

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

• Marc

8 years ago

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

• AUTHOR

t.elia

8 years ago from Northern Ireland

Simba,

What are you working on multiplication?? Please clarify.

Telia

• AUTHOR

t.elia

8 years ago from Northern Ireland

MAHI,

Arithematic Progression:-

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.

• MAHI

8 years ago

WHAT IS AP?

• ogleoo

8 years ago

i keep forgetting that n means term number!!!

• ogleoo

8 years ago

thank you so much

• AUTHOR

t.elia

8 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

• student

8 years ago

hello plaese help me find Recurrence Relations

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

• AUTHOR

t.elia

8 years ago from Northern Ireland

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

• Paul

8 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

• AUTHOR

t.elia

8 years ago from Northern Ireland

Hi Jamie

• Jamie

8 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?

• shreyansh

8 years ago

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

• AUTHOR

t.elia

8 years ago from Northern Ireland

Oscar,

3 x 1 = 3 + 4 = 7

3 x 2 = 6 + 4 = 10

3 x 3 = 9 + 4 = 13 etc

Good Luck & hope this helps

• oscar

8 years ago

• bubble 543

8 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

• AUTHOR

t.elia

8 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

• liv

8 years ago

26-4n?

• Mahmoud Shahoud

8 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

• AUTHOR

t.elia

8 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

• AUTHOR

t.elia

8 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

• AUTHOR

t.elia

8 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

• David

8 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.

• dom

8 years ago

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

• AUTHOR

t.elia

8 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

• Me

8 years ago

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

• Russ

8 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!

• Bex

8 years ago

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

Doing my GCSEs and stuck on homework.

• renee

8 years ago

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

• saf

8 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

working