Chapter 13, Problem 10CR

To determine

To find: The ${33}^{rd}$ form in the sequence and the sum of the first $20$ terms.

Answer to Problem 10CR

The ${33}^{rd}$ term in the sequence is $125$ and the sum of the first $20$ terms is $700$ .

Explanation of Solution

Given function: The sequence $-3,1,5,9,\mathrm{.....}$

Calculation:

The sequence $-3,1,5,9,\mathrm{.....}$

The common difference is,

  $\begin{array}{l}d=1-\left(-3\right)\\ =1+3\\ =4\end{array}$

First term, $a_1=-3$ .

Let’s, find the ${33}^{rd}$ term,

  $\begin{array}{l}{a}_{33}=a_1+\left(n-1\right)d\\ =-3+\left(33-1\right)\times 4\\ =-3+32\times 4\\ =-3+128\\ =125\end{array}$

Let’s find the sum of the find $20$ terms,

  $a_1=-3,$

  $\begin{array}{l}{a}_{20}=-3+\left(20-1\right)\times 4\\ =-3+19\times 4\\ =73\end{array}$

Therefore,

  $\begin{array}{l}\text{The sum}=n\left(\frac{a_1+a_n}{2}\right)\\ =20\left(\frac{-3+73}{2}\right)\\ =20\times 35\\ =700\end{array}$

Hence, the ${33}^{rd}$ term in the sequence is $125$ and the sum of the first $20$ terms is $700$ .