Find the sum of the series whose n^ th term is: n(n + 1)(n + 4) - Vidyadip

Question

Find the sum of the series whose $n^{th}$ term is: $n(n + 1)(n + 4)$

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Answer

$T_n = n(n + 1)(n + 4)$
$T_n = (n^2 + n)(n + 4)$
$T_n = n^3 + 5n^2 + 4n$
Let $S_n$ be the sum of n terms of the given series.
Now,
$\text{S}_\text{n}=\sum\limits^{\text{n}}_{\text{k}=1}\text{T}_\text{k}$
$\Rightarrow\text{S}_\text{n}=\sum\limits^{\text{n}}_{\text{k}=1}(\text{k}^3+5\text{k}^2+4\text{k})$
$\Rightarrow\text{S}_\text{n}=\sum\limits^{\text{n}}_{\text{k}=1}\text{k}^3+5\sum\limits^{\text{n}}_{\text{k}=1}\text{k}^2+4\sum\limits^{\text{n}}_{\text{k}=1}\text{k}$
$\Rightarrow\text{S}_\text{n}=\frac{\text{n}^2(\text{n}+1)^2}{4}+\frac{5\text{n}(\text{n}+1)(2\text{n}+1)}{6}+\frac{4\text{n}(\text{n}+1)}{2}$
$\Rightarrow\text{S}_\text{n}=\frac{\text{n}(\text{n}+1)}{2}+\Big[\frac{\text{n}(\text{n}+1)}{2}+\frac{5(2\text{n}+1)}{3}+4\Big]$
$\Rightarrow\text{S}_\text{n}=\frac{\text{n}(\text{n}+1)}{2}\big[3\text{n}(\text{n}+1)+10(2\text{n}+1)+24\big]$
$\Rightarrow\text{S}_\text{n}=\frac{\text{n}(\text{n}+1)}{2}\big(3\text{n}^2+23\text{n}+34)$

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