Question

5²ⁿ⁺² −24n −25 is divisible by 576 for all n ∈ N.

 

Answer 1

Let P(n) be the given statement.
Now,

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