In an A.P. prove that $t_{m + n} + t_{m - n} = 2 t_{m}$ .

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Solution

Any term $t_{n}$ of AP = a+(n-1)d
$∴ t_{m + n} = a + (m + n - 1) d$
$t_{m - n} = a + (m - n - 1) d$
$∴ t_{m + n} + t_{m - n} = a + m d + / n d - d + a + m d - / n d - d$
=2a+2md-2d
=2(a+(m-1)d)
$= 2 t_{m}$

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t_{m + n} + t_{m - n} = 2 t_{m}

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