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प्रश्न
In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.
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उत्तर
Given:
\[a_{24} = 2 a_{10} \]
\[ \Rightarrow a + \left( 24 - 1 \right)d = 2\left[ a + \left( 10 - 1 \right)d \right]\]
\[ \Rightarrow a + 23d = 2(a + 9d)\]
\[ \Rightarrow a + 23d = 2a + 18d\]
\[ \Rightarrow 5d = a . . . (i)\]
\[\text { To prove }: \]
\[ a_{72} = 2 a_{34} \]
\[\text { LHS: } a_{72} = a + \left( 72 - 1 \right)d\]
\[ \Rightarrow a_{72} = a + 71d\]
\[ \Rightarrow a_{72} = 5d + 71d \left( \text { From }(i) \right)\]
\[ \Rightarrow a_{72} = 76d\]
\[\text { RHS }: 2 a_{34} = 2\left[ a + \left( 34 - 1 \right)d \right]\]
\[ \Rightarrow 2 a_{34} = 2\left( a + 33d \right)\]
\[ \Rightarrow 2 a_{34} = 2(5d + 33d) \left( \text { Form }(i) \right)\]
\[ \Rightarrow 2 a_{34} = 2\left( 38d \right)\]
\[ \Rightarrow 2 a_{34} = 76d\]
∴ RHS = LHS
Hence, proved.
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