Question

3²ⁿ⁺² −8n − 9 is divisible by 8 for all n ∈ N.

Answer 1

Let P(n) be the given statement.
Now,

\[P(n): 5^{2n + 2} - 24n - 25 \text{ is divisible by 576 for all }  n \in N . \]
\[\text{ Step } 1: \]
\[P(1) = 5^{2 + 2} - 24 - 25 = 625 - 49 = 576 \]
\[\text{ It is divisible by }  576 . \]
\[\text{ Thus, P(1) is true}  . \]
\[\text{ Step2:}  \]
\[\text{ Let P(m) be true . } \]
\[Then, \]
\[ 5^{2m + 2} - 24m - 25 \text{ is divisible by } 576 . \]
\[\text { Let } 5^{2m + 2} - 24m - 25 = 576\lambda, \text{ where } \lambda \in N . \]
\[\text { We need to show that P(m + 1) is true whenever P(m) is true }  . \]
\[ \text{ Now, } \]
\[P(m + 1) = 5^{2m + 4} - 24(m + 1) - 25\]
\[ = 5^2 \times (576\lambda + 24m + 25) - 24m - 49\]
\[ = 25 \times 576\lambda + 600m + 625 - 24m - 49\]
\[ = 25 \times 576\lambda + 576m + 576\]
\[ = 576(25\lambda + m + 1) \]
\[\text{ It is divisible by } 576 . \]
\[\text{ Thus, P(m + 1) is true }  . \]
\[\text{ By the principle of mathematical induction, P(n) is true for all n }  \in N . \]