Let S n denote the sum of first n terms of an arithmetic progression. If S 20 = 790 and S 10 = 145 , then S 15 - S 5 is [2024]

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Solution

Q.
Let $S_{n}$ denote the sum of first $n$ terms of an arithmetic progression. If $S_{20} = 790$ and $S_{10} = 145,$ then $S_{15} - S_{5}$ is [2024]

1 390

2 405

3 410

4 395

Ans.
(4)

Given, $S_{20} = 790$

$\Rightarrow \frac{20}{2} (2 a + 19 d) = 790 \Rightarrow 2 a + 19 d = 79$ ...(i)

Given, $S_{10} = 145$

$\frac{10}{2} (2 a + 9 d) = 145 \Rightarrow 2 a + 9 d = 29$ ...(ii)

From (i) and (ii), we get $10 d = 50 \Rightarrow d = 5$

From (i), we get $a = \frac{79 - 95}{2} = - 8$

So, $S_{15} - S_{5} = \frac{15}{2} [- 16 + 70] - \frac{5}{2} [- 16 + 20] = 395$

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