Chapter 9.1, Problem 58E

To determine

To Explain: The exponents of the terms of an arithmetic sequence from a geometric sequence

Explanation of Solution

Given Information: $\left\{a_n\right\}$ is an arithmetic sequence

Concept Used:

 • The $n^{th}$ term of an arithmetic sequence with first term $a$ and common difference $d$ is $a_n=a+(n-1)d$

 • $x^{p+q}=x^p\cdot x^q$

 • $x^{p-q}=\frac{x^p}{x^q}$

The $n^{th}$ term and the $n+1^{th}$ of the arithmetic series are

  $a_n=a+(n-1)d$

  $a_{n+1}=a+nd$

Then the exponents of the terms are

  ${10}^{a_n}={10}^{a+(n-1)d}={10}^a\cdot {10}^{(n-1)d}$

  ${10}^{a_{n+1}}={10}^{a+nd}={10}^a\cdot {10}^{nd}$

The ratio of the exponent of the two terms is therefore

  $\frac{{10}^{a_{n+1}}}{{10}^{a_n}}=\frac{{10}^a\cdot {10}^{nd}}{{10}^a\cdot {10}^{(n-1)d}}={10}^{nd-(n-1)d}={10}^d$

It means that the ratio of exponents of any two consecutive terms of the arithmetic sequence is constant and is equal to ${10}^d$ .

Therefore, the sequence of exponents of the terms of the arithmetic sequence form a geometric sequence, with common ratio ${10}^d$ .