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  • If the 9thterm of an AP is zero, prove that its 29thterm is twice its 19thterm.

If the 9th term of an AP is zero, prove that its 29th term is twice its 19th term.

Answers (1)

Let the first term = a  

common difference = d  

Given:- a\textsubscript{9} = 0\\

\Rightarrow \therefore a + 8d = 0 \quad (\text{using } a_{n} = a + (n-1)d) \\ \Rightarrow \therefore a = -8d \quad \ldots (1) \\ \text{To prove: } 2a_{19} = a_{29} \\ a_{19} = a + 18d \quad (\text{using } a_{n} = a + (n-1)d) \\ = -8d + 18d \quad \{ a = -8d \text{ from equation (1)} \} \\ = 10d \quad \ldots (2) \\ a_{29} = a + (29 - 1)d \\ = a + 28d \\ = -8d + 28d \quad \{ a = -8d \text{ from equation (1)} \} \\ = 20d \\ = 2 \times 10d \\ \text{By equation (2)} \\ a_{29} = 2a_{19}

Hence proved

Posted by

infoexpert21

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