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If Sn denotes the sum of first n terms of an AP,prove that S12=3 (S8_S4)

Posted by Nainu Choudhary 6 years, 5 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 6 years, 5 months ago

Let a be the first term and d be the common difference of the given AP. Then,
S_n= {tex}\frac{n}{2}{/tex}{tex} \cdot {/tex}[2a+(n-l)d],
{tex}\therefore{/tex} {tex}3(S_8-S_4) = 3{/tex}[{tex}\frac{8}{2}{/tex}{tex}(2a+7d)-{/tex}{tex}\frac{4}{2}{/tex}{tex}(2a+3d)]{/tex}
= {tex}3[4(2a+7d)- 2(2a+3d)] = 6(2a+11d){/tex}
{tex}= \frac { 12 } { 2 } \cdot ( 2 a + 11 d ) = S _ { 12 }{/tex}.
Hence, S₁₂= 3(S₈-S₄).

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