Let S _ a denote the sum of first n terms an arithmetic...

Let

{S}_{a}

denote the sum of first

n

terms an arithmetic progression. If

{S}_{20} = 790

and

{S}_{10} = 145

, then

{S}_{15} -

{S}_{5}

is :

Options

A $395$
B $390$
C $405$
D $410$

Correct answer

A. $395$

Step-by-step solution

Given,

In an arithmetic progression sum of first $20$ terms is $790$

So, let $a$ be the first term and $d$ be common difference,

So, ${S}_{20} = \dfrac{20}{2}\left[2a + 19d\right] = 790 . . . . . \left(1\right)$

And sum of the first $10$ terms is $145$

So, ${S}_{10} = \dfrac{10}{2}\left[2a + 9d\right] = 145 . . . . . \left(2\right)$

By solving equation \left(1\right) \text{ and } \left(2\right) $we get,$ a = - 8 \text{ and } d = 5 $ Then,$ {S}_{15} - {S}_{5} = \dfrac{15}{2}\left[2 \times \left(- 8\right) + 14 \times 5\right] - \dfrac{5}{2}\left[2 \times \left(- 8\right) + 4 \times 5\right] $$ ⇒ {S}_{15} - {S}_{5} = 405 - 10 = 395$

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