Let S n denote the sum of the first n terms of an arithmetic progression. If S 10 = 390 and the ratio of the tenth and the fifth terms is 15 : 7, then S 15 - S 5 is equal to: [2024]

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Solution

Q.
Let $S_{n}$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10} = 390$ and the ratio of the tenth and the fifth terms is 15 : 7, then $S_{15} - S_{5}$ is equal to: [2024]

1 800

2 890

3 790

4 690

Ans.
(3)

We have, $S_{10} = 390$

$\Rightarrow 5 (2 a + 9 d) = 390 \Rightarrow 2 a + 9 d = 78$ ...(i)

Also, $\frac{a_{10}}{a_{5}} = \frac{15}{7} \Rightarrow \frac{a + 9 d}{a + 4 d} = \frac{15}{7}$

$\Rightarrow 7 a + 63 d = 15 a + 60 d \Rightarrow 8 a = 3 d$

$\Rightarrow a = \frac{3}{8} d$ ...(ii)

From (i) & (ii), we get $d = 8$ and $a = 3$

Now, $S_{15} - S_{5} = \frac{15}{2} [6 + 112] - \frac{5}{2} [6 + 32] = 790$

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