Chapter 8.2, Problem 61E

To determine

To calculate: The sum of the positive odd integers less than 300.

Answer to Problem 61E

The sum is $22,500$ .

Explanation of Solution

Given information:

The sum of positive odd integers.

Formula used:

Arithmetic difference − the difference of consecutive term is constant. The constant difference is called the common difference and is denoted by d.

Therefore,

  $d=a_{n+1}-a_n;n\ge 1$ .

The $nth$ tem of an arithmetic sequence with first term $a_1$ and common difference $d$ is given by : $a_n=a_1+(n-1)d$ .

The sum of the first n terms of arithmetic series is

  $S_n=n\left(\frac{a_1+a_n}{2}\right).$

Calculation:

Consider ,

The first odd term will be 1.

The last odd term will be 299.

On adding 1 to the last odd term we get,

  $=1+299=300$

Since, 300 is even term. This implies

  $\begin{array}{l}=\frac{300}{2}\\ =150\end{array}$

This is the average of both even terms and odd terms.

Therefore, the number of odd terms arte 150.

The $nth$ term for odd series will be $2n-1$ .

  ${\displaystyle \sum _{i=1}^{150}2i-1}$

Now, the sum of 150 odd terms are

  $\begin{array}{l}{a}_1=1\\ a_{150}=299\end{array}$

The sum of 150 odd terms is −

  $\begin{array}{l}{S}_{150}=150\left(\frac{1+299}{2}\right)\\ =150\left(\frac{300}{2}\right)\\ =150\left(150\right)\\ =22,500\end{array}$

Therefore , the sum is $22,500$ .