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How To Solve Arithmetic Progression Without Using Formula

Updated on August 15, 2019
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Arithmetic progression

In general, there are two types of Progression :

Arithmetic Progression (A.P.) and Geometric Progression (G.P).

To identify the Arithmetic progression, make sure all the numbers in the sequence are separated by a Common Difference. (T2 - T1= T3 - T2)

To identify the Geometric progression, make sure all the numbers in the sequence are separated by a Common Ratio. (T2 / T1 = T3 / T2)

There is a specific term (Tn) and the sum of terms (Sn) in both Arithmetic and Geometric progression. Normally, we determine the Tn and Sn using the Arithmetic and Geometric progression formula. Of course, sometimes we do not need these formulae in order to get Tn and Sn.

To solve arithmetic progression or arithmetic sequence without using a formula. First, we need to determine the given sequence is an arithmetic progression by + or - with a constant number. For example :

Set A = 1, 3, 5, 7, 9 is arithmetic progression because the number in this sequence is constantly +2.

Set B = 1, 3, 4, 7, 11 is not an arithmetic sequence because numbers in this sequence do not have the common difference. (e.g. 3 - 1 = 2 , 4 - 3 = 1, 7 - 4 = 3).

Problem Solving (Without Formula).

You are given a set of Arithmetic Progression (A.P.)

3, 6, 9, 12, .....

1. Determine the 10th number in this sequence.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30

Solution: 30

2. Find the value of S10.

3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 + 30

Solution : 165

3. Find the sum of T6 until T11.

18 + 21 + 24 + 27 + 30 + 33

Solution : 153

4. Given the number 57 is one of the number in this sequence. Determine the value of Tn.

T10 = 30,

30 (T10), 33, 36, 39, 42, 45, 48, 51, 54, 57(T19)

Solution : 57 = T19

5. Find the sum of first seventh terms after the third terms.

3, 6, 9 (T3) , [12, 15, 18, 21, 24, 27, 30, 33] - > first seventh terms after third terms. { 12 + 15 + 18 + 21 + 24 + 27 + 30 + 33}

Solution : 180


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    • verdict profile image

      George Dimitriadis 

      7 months ago from Templestowe


      You have clearly distinguished between the two types of progression and provided some useful examples to challenge the reader.


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