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The sum of n,2n,3n term of …

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The sum of n,2n,3n term of an AP are S1,S2,S3 respectively . Prove that S3=3(S2_S1)

Posted by Bhushan Nikhade 6 years, 4 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 6 years, 4 months ago

{tex}{S_1} = \frac{n}{2}\left[ {2a + (n - 1)d} \right]{/tex}
{tex}{S_2} = \frac{{2n}}{2}\left[ {2a + (2n - 1)d} \right]{/tex}
{tex}{S_3} = \frac{{3n}}{2}\left[ {2a + (3n - 1)d} \right]{/tex}
R.H.S = 3(S₂ - S₁)
{tex} = 3\left[ {\frac{{2n}}{2}(2a + (2n - 1)d - \frac{n}{2}(2a + (n - 1)d)} \right]{/tex}
{tex} = 3\left[ {\frac{n}{2}\left[ {4a + 4nd - 2d - 2a - nd + d} \right]} \right]{/tex}
{tex} = 3\left[ {\frac{n}{2}(2a + 3nd - d)} \right]{/tex}
{tex} = \frac{{3n}}{2}\left[ {2a + (3n - 1)d} \right] = {S_3}{/tex}

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