For all n N, 72n − 48n−1 is divisible by: - Vidyadip

MCQ

For all $n\in N, 72n − 48n−1$ is divisible by:

A
$50$
✓
$2304$
C
$1234$
D
$44$

✓

Answer

Correct option: B.

$2304$

Concepts:
Suppose there is a given statement $P (n)$ involving the natural number $n$ such that
The statement is true for $n = 1,$
i.e., $P (1)$ is true, and
If the statement is true for $n = k ($where $k$ is some positive integer$)$, then the statement is also true for $n = k + 1,$ i.e., truth of $P (k)$ implies the truth of $P (k + 1).$
Then, $P (n)$ is true for all natural numbers $n$
Calculation:
Given:
$P(n) = 72n − 48n−1$
Put, $n = 1$
$P(1) = 72 − 48 \times 1 −1 = 0$
Check the expression $P(n)$ for $n = k ($where $k$ is some positive integer$) = 2, 3, 4......$
$P(2) = 7^{2n}− 48n − 1$
$= 7^4− 48 \times 2 − 1$
$= 2401 – 96 – 1$
$= 2401 – 97$
$= 2304$
$P(3) = 7^{2n}− 48n − 1$
$= 7^6− 48 \times 3 − 1$
$= 117649 – 144 – 1$
$= 117649 – 145$
$= 117504$
$= 2304 \times 51$
Since, all these numbers are divisible by $2304$ for $n = 1$ and $k = 2, 3,…..$
So, the given number is divisible by $2304$

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