Chapter 6.6, Problem 10E

To determine

To find:Whether the sequence $-1,4,-7,\mathrm{10...}$ is arithmetic, geometric or neither.

Answer to Problem 10E

The sequence $-1,4,-7,\mathrm{10...}$ is neither arithmetic nor geometric.

Explanation of Solution

Given information:The sequence $-1,4,-7,\mathrm{10...}$

Calculation:A sequence is said to be arithmetic sequence if every term of the sequence, after the first term, is obtained by adding a constant to the preceding term. The constant is known as common difference.

A sequence is said to be geometric sequence if every term of the sequence, after the first term, is obtained by multiplying a constant to the preceding term. The constant is known as common ratio.

Here, the 1^st term $=-1$

Difference between 1^st and 2^nd term $=4-(-1)=4+1=5$

Difference between 2^nd and 3^rd term $=-7-4=-11$

Difference between 3^rd and 4^th term $=10-(-7)=10+7=17$

Since, each term of the sequence is not obtained by adding a constant to the preceding term, the sequence is not arithmetic.

Again, the 1^st term $=-1$

Dividing 2^nd term by 1^st term $=\frac{4}{-1}=-4$

Dividing 3^rd term by 2^nd term $=\frac{-7}{4}=-\frac{7}{4}$

Dividing 4^th term by 3^rd term $=\frac{10}{-7}=-\frac{10}{7}$

Since, each term of the sequence is not obtained by multiplying a constant to the preceding term, the sequence is not geometric.