If the sum of first n, …
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Sia ? 6 years, 4 months ago
Let a be the first term and d be the common difference of the given AP. Then,
S1 = sum of first n terms of the given AP,
S2 = sum of first 2n terms of the given AP,
S3 = sum of first 3n terms of the given AP.
S1 ={tex}\frac{n}{2}{/tex}{tex}\cdot{/tex}{2a+(n-1)d}, S2={tex}\frac{{2n}}{2}{/tex}{tex}\cdot{/tex}{2a+(2n-1)d}, and S3= {tex}\frac{{3n}}{2}{/tex}{tex}\cdot{/tex}{2a+(3n-1)d}
{tex}\Rightarrow{/tex}3(S2-S1) = 3{tex}\cdot{/tex}[{2na+n(2n - 1)d} - {na+{tex}\frac{1}{2}{/tex}n(n-1)d}]
= 3{tex}\cdot{/tex}[na + {tex}\frac{3}{2}{/tex}n2d-{tex}\frac{1}{2}{/tex}nd] = {tex}\frac{{3n}}{2}{/tex}{tex}\cdot{/tex}[2a+3nd - d]
= {tex}\frac{{3n}}{2}{/tex}{tex}\cdot{/tex}[2a+(3n-1)d}=S3.
Hence, S3=3(S2-S1).
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