CAT 2023 Slot 3 QA Question 7: Let n be any natural number such that $5^{n - 1}

Question

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

Accepted Answer

First find the natural numbers n that satisfy the given condition.

$5^{n - 1} 
$\dfrac{5^n}{5} 
$\left(\dfrac{5}{3}\right)^n

Checking powers, this holds for $n = 1, 2, 3, 4, 5$, but not for $n = 6$.

Now find the least m required for each of these n values.

$n = 1:$ $3^2 
$n = 2:$ $3^3 
$n = 3:$

Let n be any natural number such that 5n−1<3n+15^{n - 1} < 3^{n + 1}5n−1<3n+1. Then, the least integer value of m that satisfies 3n+1<2n+m3^{n + 1} < 2^{n + m}3n+1<2n+m for each such n, is

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