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Sum of n Terms

If Tm,Tm + ...

Question

If $T_{m,} T_{m + n}$ and $T_{m - n}$ are respectively the $m t h, (m + n) t h$ and $(m - n) t h$ terms of an AP, then prove that $T_{m + n} + T_{m - n} = 2 T_{m}$

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Solution

Given $T_{n} = a + (m - 1) d$

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

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

Now $T_{m + n} + T_{m - n}$

$= a + (m + n - 1) d + a + (m - n - 1) d$

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

$= 2 a + 2 m d - d$

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

$\Rightarrow$ Hence Proved

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