Chapter 12.2, Problem 1CFU

To determine

To compare and contrast arithmetic and geometric sequences.

Explanation of Solution

Given:

The arithmetic and geometric sequences.

Calculation:

Compare

The arithmetic sequence is a sequence of terms, whose differences of successive terms is constant.

The general term of the arithmetic sequence having first term $a$ and common difference $d$ is

  $a_n=a+(n-1)d$ …… (1)

The sum of first terms of arithmetic sequence having first term $a$ and common difference $d$ is

  $S_n=\frac{n}{2}\left[2a+(n-1)d\right]$ ...... (2)

The geometric sequence is sequence of terms, whose ratio of successive terms is constant

The general term of the geometric sequence having first term $a$ and common ratio $r$ is

  $a_n=a{r}^{n-1}$ …… (3)

The sum of first terms of geometric series is

  $S_n=\frac{a(r^n-1)}{r-1}$ ...... (4)

In Arithmetic sequence the terms are continuously positive or negative, but in the case of geometric sequence the terms are either positive or negative or alternatively positive and negative

The sum of infinite arithmetic sequence is not finite, but the sum of infinite geometric sequence whose common ration less than 1 is finite

The general term of the arithmetic sequence is linear, general term of geometric sequence is exponential.