Question 17 and 18, Exercise 4.7
Solutions of Question 17 and 18 of Exercise 4.7 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.
Question 17
Sum the series up to $n$ term: $2^{2}+5^{2}+8^{2}+\ldots$
Solution.
Rough Work
Take $2+5+8+\ldots$.This is A.P with kth term $a_k=2+(k-1)(3)=2+3k-3=3k-1$.
Now consider this to make kth term of given series by just taking square.
Consider $T_k$ represents the $k$th term of the sereies, then \begin{align*}T_k&=(3k-1)^2 \\ &=9k^2-6k+1. \end{align*}
Taking summation, we have \begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (9k^{2} - 6k + 1)\\ & = 9\sum_{k=1}^{n} k^{2} -6 \sum_{k=1}^{n} k + \sum_{k=1}^{n} 1 \\ & = 9\left( \frac{n(n+1)(2n+1)}{6} \right) - 6\left( \frac{n(n+1)}{2} \right) + n \\ & = \frac{3n(n+1)(2n+1)}{2} -3 n(n+1) + n \\ & = \frac{n}{2}\left[3(n+1)(2n+1)-6(n+1) + 2\right]\\ & = \frac{n}{2}\left(6n^2+6n+3n+3-6n-6+2\right) \\ & = \frac{n}{2}\left( 6n^2+3n-1\right) \end{align*}
Thus, the sum of the series is
$\sum\limits_{k=1}^{n} T_{k} = \frac{n}{2}\left(6n^2+3n-1\right).$ 
Question 18
Sum the series up to $n$ term: $2^{2}+4^{2}+6^{2}+$
Solution.
Rough Work
Take $2+4+6+\ldots$.This is A.P with kth term $a_k=2+(k-1)(2)=2k$.
Now consider this to make kth term of given series by just taking square.
Consider $T_k$ represents the $k$th term of the sereies, then \begin{align*}T_k&=(2k)^2 \\ &=4k^2 \end{align*}
Taking summation, we have \begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (4k^2 )\\ & = 4 \sum_{k=1}^{n} k^2 \\ & = 4\left( \frac{n(n+1)(2n+1)}{6} \right) \\ & = \frac{2n}{3}(2n^2+2n+n+2) \\ & = \frac{2n}{3}(2n^2+3n+2) \\ \end{align*}
Thus, the sum of the series is
$\sum\limits_{k=1}^{n} T_{k} = \dfrac{2n}{3}(2n^2+3n+2).$ 
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