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Q.
$P(n) : 2.7^n + 3.5^n - 5$ is divisible by
Principle of Mathematical Induction
Solution:
$P(n) : 2.7^n + 3.5^n - 5$ is divisible by $24.$
For $n = 1,$
$P(1) : 2.7 + 3.5 - 5 = 24$, which is divisible by $24.$
Assume that $P(k)$ is true,
$i.e. 2.7^{k}+3.5^{k+1}-5=24q$, where $q \in N\,...\left(i\right)$
Now, we wish to prove that $P(k + 1)$ is true whenever $P(k)$ is true, i.e. $2.7^{k + 1} + 3.5^{k + 1} - 5$ is divisible by $24$. We have,
$2.7^{k+1} + 3.5^{k+1} - 5 = 2.7^{k} . 7^1 + 3.5^{k} . 5^{1} - 5$
$=7\left[2.7^{k}+3.5^{k}-5-3.5^{k}+5\right]+3.5^{k}.5-5$
$=7\left[24q-3.5^{k}+5\right]+15.5^{k}-5
=\left(7\times24q\right)-21.5^{k}+35+15.5^{k}-5$
$=\left(7\times24q\right)-6.5^{k}+30=\left(7\times24q\right)-6\left(5^{k}-5\right)
=\left(7\times24q\right)-6\left(4p\right) $ [$\because (5^k - 5)$ is a multiple of $4$]
$= (7 × 24q) - 24p = 24(7q - p)$
$= 24 × r; r = 7q - p$, is some natural number $... (ii)$
Thus, $P(k + 1)$ is true whenever $P(k)$ is true.
Hence, by the principle of mathematical induction, $P(n)$ is true for all $n\in N.$