7 n − 3 n is always divisible by Options: A . &nbsp 6 B . &nbsp 10 C . &nbsp 4 D . &nbsp 8

7n−3n is always divisible by

Discussion Forum : Principle Of Mathematical Induction

Question -

$7^{n} - 3^{n}$ is always divisible by

Options:

A . 6

B . 10

C . 4

D . 8

Answer: Option C
:
C

Substituting n=1, we get
$7^{1} - 3^{1} = 4$
Let $P (n) : 7^{n} - 3^{n} is divisible by 4$
P(1) is true
Assume P(k) is true
$7^{k} - 3^{k} is divisible by 4 7^{k} - 3^{k} = 4 m N o w, 7^{k + 1} - 3^{k + 1} = 7. (7^{k} - 3^{k}) + {4.3}^{k} = 4. (7 m + 3^{k}) \to divisible by 4$
P(k+1) is true
Hence, P(n) is true.

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Discussion Forum : Principle Of Mathematical Induction

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