CAT 2023 Slot 3 Solved Paper

Question : 49 of 66

Marks: +1, -0

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

Solution:

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=1,2,34

, 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 2 .
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 .

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