For every positive integer n, 7n – 3n is divisible by - Vidyadip

MCQ

For every positive integer $n, 7n – 3n$ is divisible by

A
$2$
✓
$4$
C
$5$
D
$6$

✓

Answer

Correct option: B.

$4$

Concept:
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:
We have to find $7^n- 3^n$ is divisible by which number
Consider $P(n): 7n - 3n$
$P (1): 7^1- 3^1= 4$
Thus, $7n - 3n$ is divisible by $4$
Let $P(k)$ is true for $n = K$
$\Rightarrow 7^k− 3^k$ is divisible by $4$
So, $7n – 3n = 4d$
Now, prove that $P(k+1)$ is true.
$\Rightarrow 7^{(k+1)}-3^{(k+1)}=7^{(k+1)}-7.3^k+7.3^k-3^{(k+1)} $
$ =7\left(7^k-3^k\right)+(7-3) 3^k $
$ =7(4 d)+(7-3) 3^k $
$ =7(4 d)+4.3^k $
$ =4\left(7 d+3^k\right) $
Hence, $P (n): 7^n- 3^n$ is divisible by $4$ is true.

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