Question

Use mathematical induction to prove that 4 is a factor of 9n − 5
n
for all integers n ≥ 1.

Answer 1

Putting n=1

We get

$9-5=4$

Which is divisible by 4 so 4 is the factor of 4

Let

$T(k) =9k-5k=4m\ for \ some \ m$ is true $\ \ \ \ ....(1)$

For

$T(k+1)$

Now,

$$T(k+1)= 9(k+1)-5(k+1)$$

$$=9k+9-5k-5=4m+4$$

$$=4(m+1)$$

Thus, $T(k+1)$ is divisible by 4, hence 4 is a factor of it. By the principle of mathematical induction, 4 is a factor of 9n-5n for all integers $n \geq 1.$