cat 2023 Question 7

Question

Let n be any natural number such that $5^{n-1} \lt 3^{n+1}$. Then, the least integer value of m that satisfies $3^{n+1} \lt 2^{n+m}$ for each such n, is

Accepted Answer

It is given that $5^{n−1}$ < $3^{n+1}$, where n is a natural number. By inspection, we can say that the inequality holds when n = 1, 2, 3, 4, and 5.

Now, we need to find the least integer value of m that satisfies $3^{n+1}$ < $2^{n+m}$

For, n =1, the least integer value of m is 3.

For, n = 2, the least integer value of m is 3

For, n = 3, the least integer value of m is 4.

For, n = 4, the least integer value of m is 4.

For, n = 5, the least integer value of m is 5.

Hence, the least integer value of m such that for all the values of n, the equation holds is 5.3

Question 7.

Question Explanation