Question

The sum of first 7term of an



a.P is 49 and that of first 17 terms of it is 289 .Find the sum of first n terms

Answer 1

The sum of first 7 terms of AP is 49

This means that :-

$\frac{7}{2}(2a + 6d) = 49 \\ \\ \\ 2a + 6d = \frac{49 \times 2}{7} = 14 \\ \\ \\ a + 3d = \frac{14}{2} = 7$

$\rule{200}{2}$

The sum of first 17 terms of AP is 289

This implies :-

$\frac{17}{2}(2a + 16d) = 289 \\ \\ \\ 2a + 16d = \frac{289 \times 2}{17} = 17 \times 2 \\ \\ \\ a + 8d = 17$

$\rule{200}{2}$

Hence,

We have two equations :-

Solving the equations :-

$a + 8d = 17 \\ a + 3d = 07 \\ - \: \: - \: \: \: \: \:- \\ = = = = = = = = = = \\ 5d = 10 \\ \\ d = \frac{10}{5} = 2 \\ \\ \\ \\ a + 3(2) = 7 \\ \\ a = 7 - 6 = 1$

$\rule{200}{2}$

Now we have a =7 and d =2

Therefore,

Sum of n terms of AP will be :-

$= \frac{n}{2}(2a + (n - 1)d) \\ \\ \\ = \frac{n}{2}(2 + (n - 1)2) \\ \\ \\ = \frac{n}{2}(2 + 2n - 2) \\ \\ \\ = \frac{n}{2}(2n) \\ \\ \\ = {n}^{2}$

$\rule{200}{2}$