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Divisibility by 3

Prove that ...

Question

Prove that $t_{m + n} + t_{m - n} = 2 t_{m}$ , where $t_{m}$ is the $n^{t h}$ term of AP

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Solution

Formula,

$t_{n} = a + (n - 1) d$

$t_{m + n} = a + [(m + n) - 1] d$

$t_{m - n} = a + [(m - n) - 1] d$

$t_{m + n} - t_{m - n} = a + [(m + n) - 1] d - a - [(m - n) - 1] d$

$= 2 a + 2 m d - 2 d$

$= 2 [a + (m - 1) d]$

$= 2 t_{m}$

Hence proved.

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