Chapter 8.2, Problem 59E

To determine

To show: The sum of a series doubles when the common difference of an arithmetic series is doubled and the first term and number of terms in the series remain unchanged.

Answer to Problem 59E

The statement is incorrect.

Explanation of Solution

Given information:

The sum of a series doubles when the common difference of an arithmetic series is doubled and the first term and number of terms in the series remain unchanged

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 ,

  $a_n=a_1+(n-1)d$

On doubling the common difference we get,

  $a_n=a_1+(n-1)2d$

Now,

This shows that both the last term have different terms. Hence, on doubling the common difference , the terms also gets change.