Chapter 9.4, Problem 55HP

To determine

To explain: Whether the mentioned fraction $\frac{1}{2^n}$ increase or decrease.

Answer to Problem 55HP

The mentioned fraction $\frac{1}{2^n}$ decrease as the value of n increase.

Explanation of Solution

Given information:

The expression mentioned in the problem is $\frac{1}{2^n}$ .

Formula used:

If the denominator part is greater than numerator part than value of the fraction will always decrease as increase its power.

Calculation:

Consider the statement that the expression mentioned in the problem is $\frac{1}{2^n}$ .

Recall that if the denominator part is greater than numerator part than value of the fraction will always decrease as increase its power.

Let us understand with example:-

Let $n=1$ then the mentioned fraction will be $\frac{1}{2}=0.5$ .

Let $n=2$ then the mentioned fraction will be $\frac{1}{2^2}=\frac{1}{4}=0.25$ .

Let $n=3$ then the mentioned fraction will be $\frac{1}{2^3}=\frac{1}{8}=0.125$ .

Similarly on moving further like this,

Let $n=10$ then the mentioned fraction will be $\frac{1}{2^{10}}=\frac{1}{1024}\approx 0.00097$ .

Therefore we observe that as we increase the value of n the function value will always decrease.