In an A.P., show that a_ m+n +a_ m-n =2a_m. - Vidyadip

Question

In an A.P., show that $\text{a}_{\text{m}+\text{n}}+\text{a}_{\text{m}-\text{n}}=2\text{a}_\text{m}.$

✓

Answer

It is given that the sequence $<\text{a}_\text{n}>$ is an A.P.
$\therefore\text{a}_\text{n}=\text{a}+(\text{n}-1)\text{d}\ .....(1)$
Similarly from (1)
$\text{a}_{\text{m+n}}=\text{a}+(\text{m}+\text{n}-1)\text{d}\ .....(2)$
$\text{a}_{\text{m+n}}=\text{a}+(\text{m}-\text{n}-1)\text{d}\ .....(3)$
Adding (2) and (3)
$\text{a}_{\text{m+n}}+\text{a}_{\text{m}-\text{n}}=(\text{a}+(\text{m+n}-1)\text{d})(\text{a}+(\text{m}-\text{n}-1)\text{d})$
$=2\text{a}+(\text{m+n}-1+\text{m}-\text{n}-1)\text{d}$
$=\text{2a}+\text{2d}(\text{m}-1)$
$=2(\text{a}+(\text{m}-1)\text{d})$
$=\text{2a}_\text{m}$
Hence proved.

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