Write the sum of the series 2 + 4 + 6 + 8 + ... + 2n. - Vidyadip

Question

Write the sum of the series $2 + 4 + 6 + 8 + ... + 2n$.

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Answer

Let $T_n$ be the term of the given series and $S_n$ be the sum of the given series.
$\therefore\ \text{T}_\text{n}=2\text{n}$
$\therefore\ \text{S}_\text{n}=\sum\limits^{\text{n}}_{\text{k}=1}\text{T}_\text{k}=\sum\limits^{\text{n}}_{\text{k}=1}2\text{k}$
$=2\sum\limits^{\text{n}}_{\text{k}=1}\text{k}$
$=2\Big[\frac{\text{n}(\text{n}+1)}{2}\Big]$
$=\text{n}(\text{n}+1)$
Hence, $\text{S}_\text{n}=\text{n}(\text{n}+1)$

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