Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSArithmetic Progressions3 Marks
Question
In an A.P., show that $\text{a}_{\text{m}+\text{n}}+\text{a}_{\text{m}-\text{n}}=2\text{a}_\text{m}.$
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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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