Chapter 7.2, Problem 53E

Solution

To identify: Whether the statement is true or false “if the common difference of an arithmetic series is doubled while the first term and number of terms in the series remain unchanged, then the sum of the series is doubled.”

This statement is false.

Given information:

If the common difference of an arithmetic series is doubled while the first term and number of terms in the series remain unchanged, then the sum of the series is doubled.

Explanation:

If the difference between every pair of consecutive terms in a sequence is the same is called the common difference.

  $d=a_1-a_2$

An arithmetic series is the sum of a sequence $\left\{a_k\right\},k=1,2,\mathrm{3.....},$ in which each term is computed from the previous one by adding (or subtracting) to $n$ term.

  $S_n=\frac{1}{2}n\left(a_1+a_n\right)$

If an arithmetic series' common difference is doubled while the first term and number of terms in the series remain unchanged, the sum of the series is not doubled because the first term is not doubled, so this statement is false.