Chapter 6.5, Problem 1E

To determine

To compare:The sequences $2,4,6,8,\mathrm{10...}$ and $2,4,8,16,\mathrm{32...}$

Answer to Problem 1E

The sequence $2,4,6,8,\mathrm{10...}$ is an arithmetic sequence.

The sequence $2,4,8,16,\mathrm{32...}$ is a geometric sequence.

Explanation of Solution

Given information:The sequences $2,4,6,8,\mathrm{10...}$ and $2,4,8,16,\mathrm{32...}$

Consider the sequence $2,4,6,8,\mathrm{10...}$

Difference between 1^st and 2^nd term $=4-2=2$

Difference between 2^nd and 3^rd term $=6-4=2$

Difference between 3^rd and 4^th term $=8-6=2$

Difference between 4^th and 5^th term $=10-8=2$

And so on.

Thus, each term of the sequence, other than the first term, is obtained by adding the constant 2 to the preceding term.

So, the sequence $2,4,6,8,\mathrm{10...}$ is an arithmetic sequence with common difference 2.

Now, consider the sequence $2,4,8,16,\mathrm{32...}$

Division of 2^nd term by 1^st term $=4÷2=2$

Division of 3^rd term by 2^nd term $=8÷4=2$

Division of 4^th term by 3^rd term $=16÷8=2$

Division of 5^th term by 4^th term $=32÷16=2$

And so on.

Thus, each term of the sequence, other than the first term, is obtained by multiplying the constant 2 to the preceding term.

So, the sequence $2,4,8,16,\mathrm{32...}$ is a geometric sequence with common ratio 2.