====== 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. **

<callout type="tip" title="Rough Work" icon="true">
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.
</callout>


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 
$\sum\limits_{k=1}^{n} T_{k} = \frac{n}{2}\left(6n^2+3n-1\right).$ GOOD



=====Question 18=====
Sum the series up to $n$ term: $2^{2}+4^{2}+6^{2}+$

** Solution. **

<callout type="tip" title="Rough Work" icon="true">
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.
</callout>


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 
$\sum\limits_{k=1}^{n} T_{k} = \dfrac{2n}{3}(2n^2+3n+2).$ GOOD m(






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