If sn denotes the sum of first n term of AP . Prove that S12=3(s8-s4)

1.	If sn denotes the sum of first n term of AP . Prove that S12=3(s8-s4)
Answer» Let a be the first term and d be the common difference of the given AP. Then,Sn= {tex}\\frac{n}{2}{/tex}{tex} \\cdot {/tex}[2a+(n-l)d],{tex}\\therefore{/tex}\xa0{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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