Question

if Sn denote the sum of first n terms of Ap prove that S30 =3(S20-S10) ​

Answer 1

Given:-

Sn denotes the sum of first n terms of AP.

To prove:-

S30 = 3 (S20 - S10)

Proof :-

Let a be the first term and d be the common difference,

We take R. H. S,

= 3 (S20 - S10)

S20 = $\bold{</strong><strong>[</strong><strong>\frac{20}{2} </strong><strong>(</strong><strong>a + 19d</strong><strong>)</strong><strong>]</strong><strong>}$

Also,

S10 = $\bold{</strong><strong> </strong><strong> </strong><strong>[</strong><strong>\frac{10}{2} </strong><strong>(</strong><strong>a + 9d</strong><strong>)</strong><strong>]</strong><strong>}$

Now,

$= 3</strong><strong>[</strong><strong> \frac{20}{2} (a + 19d) - \frac{10}{2} (a + 9d)</strong><strong>]</strong><strong>$

$= 3</strong><strong>[</strong><strong>10(a + 19d) - 5(a + 9d)</strong><strong>]</strong><strong>$

$</strong><strong> </strong><strong>=</strong><strong> </strong><strong>3</strong><strong>[</strong><strong>(10a + 190d) -( 5a + 45d)</strong><strong>]</strong><strong>$

$</strong><strong>=</strong><strong> </strong><strong>3</strong><strong>[</strong><strong>(10a - 5a + 190d - 45d)</strong><strong>]</strong><strong>$

$</strong><strong>=</strong><strong> </strong><strong>3(5a + 145d)$

$</strong><strong>=</strong><strong> </strong><strong>15a + 435d$

Now,by solving L. H. S,

= S20

$= </strong><strong>[</strong><strong> \frac{30}{2} (2a + 29d)</strong><strong>]</strong><strong>$

$= 15</strong><strong>[</strong><strong>(a + 29d)</strong><strong>]</strong><strong>$

$= </strong><strong>1</strong><strong>5</strong><strong>a + 435d$

Since,

L. H. S = R. H. S

Therefore,

$\boxed{\sf{ </strong><strong>S30</strong><strong> </strong><strong>=</strong><strong> </strong><strong>(</strong><strong> </strong><strong>S20</strong><strong> </strong><strong>-</strong><strong> </strong><strong>S10</strong><strong>)</strong><strong>}}$