Question

The 1st,2nd and 3rd terms of an arithmetic progression are 8-x,3x and 4x+1 respectively .calculate the value of x and find the sum of the first 8th terms of the progression

Answer 1

In an arithmetic progression, consecutive terms differ by a fixed constant d, so that

8 - x + d = 3x

3x + d = 4x + 1

Solving for d in both equations gives

d = 4x - 8

d = x + 1

Solve for x :

4x - 8 = x + 1

3x = 9

x = 3

Then d = 3 + 1 = 4.

So the first three terms of the sequence are 5, 9, and 13.

The nth term of the sequence is 5 + 4(n - 1) = 4n + 1, so the sum of the first 8 terms is

`\displaystyle\sum_(n=1)^84n+1=4\sum_(n=1)^8n+\sum_(n=1)^81=\frac{4\cdot8\cdot9}2+8=\boxed{152}`