Chapter 12.1, Problem 26E

To determine

Calculate the nth term of a sequence.

Answer to Problem 26E

The nth term of the sequence is $a_n={(-\frac{1}{3})}^n$ .

Explanation of Solution

Given:

It is given in the question the first few terms are $-\frac{1}{3},\frac{1}{9},-\frac{1}{27},\frac{1}{81}\mathrm{........}$ .

Concept Used:

In this, use the concept of understanding the pattern of the first few terms and make the nth sequence.

Calculation:

Here, the first few terms are $-\frac{1}{3},\frac{1}{9},-\frac{1}{27},\frac{1}{81}\mathrm{........}$ .

It is seen that, the sequence alternates between negative and positive terms.

  $a_{1}and{a}_3$ are negative, $a_{2}and{a}_4$ are positive.

The nth term is equal to $-\frac{1}{3}$ raised to the nth power. When n is even, the negative sign will disappear, resulting in the alternating signs.

So, the sequence is $a_n={(-\frac{1}{3})}^n$ .

Conclusion:

The sequence is $a_n={(-\frac{1}{3})}^n$ .