Question 19 and 20, Exercise 4.7
Solutions of Question 19 and 20 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 19
Sum the series up to $n$ term: $1^{3}+3^{3}+5^{3}+$
Solution.
Rough Work
Take $1+3+5+\ldots$.This is A.P with kth term $a_k=1+(k-1)(2)=1+2k-2=2k-1$.
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&=(2k-1) \\ &=9k^2-6k+1. \end{align*}
Taking summation, we have \begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (2k - 1)\\ & = 2 \sum_{k=1}^{n} k - \sum_{k=1}^{n} 1 \\ & = 2\left( \frac{n(n+1)}{2} \right) - n \\ & = n(n+1)= n \\ & = n^2 \end{align*}
Thus, the sum of the series is $\sum\limits_{k=1}^{n} T_{k} = n^2.$
Question 20
Sum the series up to $n$ term: $2+5+10+17+\ldots$ to $n$ terms.
Solution.
Given: \begin{align*}&2+5+10+17\ldots\\ & (1^2+1)+(2^2+1)+(3^2+1)+(4^2+1)+\ldots\\ &(k^2+1)\end{align*}
Consider $T_k$ represents the $k$th term of the sereies, then \begin{align*}T_k&=k^2+1 \end{align*}
Taking summation, we have \begin{align*}\sum_{k=1}^{n} T_{k} &= \sum_{k=1}^{n} (k^2 + 1)\\ & = \sum_{k=1}^{n} k^2 + \sum_{k=1}^{n} 1 \\ & = \frac{n(n+1)(2n+1)}{6} + n \\ & = \frac{n}{6}\left((n+1)(2n+1) + 6\right) \\ & = \frac{n}{6}\left(2n^2+3n+1 + 6\right) \\ & = \frac{n}{6}\left(2n^2+3n+7\right) \end{align*}
Thus, the sum of the series is $\sum\limits_{k=1}^{n} T_{k} = \dfrac{n}{6}\left(2n^2+3n+7\right) $
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