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If S1 Be the Sum of (2n + 1) Terms of an A.P. and S2 Be the Sum of Its Odd Terms, the Prove That: S1 : S2 = (2n + 1) : (N + 1) - Mathematics

If S1 be the sum of (2n + 1) terms of an A.P. and S2 be the sum of its odd terms, the prove that:
S1 : S2 = (2n + 1) : (n + 1)

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Solution

\[\text { Let the A . P . be  }a, a + d, a + 2d . . . \]

\[ \therefore S_1 = \frac{2n + 1}{2}\left[ 2a + (2n + 1 - 1)d \right]\]

\[ \Rightarrow S_1 = \frac{2n + 1}{2}\left[ 2a + (2n)d \right]\]

\[ \Rightarrow S_1 = (2n + 1)(a + nd) . . . (i)\]

\[ S_2 = \frac{n + 1}{2}\left[ 2a + (n + 1 - 1) \times 2d \right]\]

\[ \Rightarrow S_2 = \frac{n + 1}{2}\left[ 2a + 2nd \right]\]

\[ \Rightarrow S_2 = (n + 1)\left[ a + nd \right] . . . (ii)\]

\[\text  { From (i) and (ii), we get }: \]

\[\frac{S_1}{S_2} = \frac{2n + 1}{n + 1}\]

\[\text { Hence, proved } .\]

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 19 Arithmetic Progression
Exercise 19.4 | Q 30 | Page 31
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