CBSE BoardEnglish MediumSTD 11 ScienceMathsMathematical Induction5 Marks
Question
$5^{2n} - 1$ is divisible by 24 for all $\text{n}\in\text{N}.$
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Answer
Let $P(n)$ be the given statement.
Now,
$p(n): 5^{2 n}-1$ is divisible by 24 for all $n \in N$.
Step 1:
$p(1): 5^2-1=25-1=24$
It is divisible by 24 .
Thus, $p(1)$ is true.
Step 2:
Let $P(m)$ be true.
Then, $5^{2 m}-1$ is divisible by 24 .
Now, let $5^{2 m}-1=24 \lambda$, where $\lambda \in N$.
We need to show that $p(m+1)$ is true whenever $p(m)$ is true.
Now,
$P(m + 1) = 5^{2m+2} - 1$
$= 5^{2m}5^2 - 1$
$=25(24\lambda+1)-1$
$=600\lambda+24$
$=24(25\lambda+1)$
It is divisible by 24.
Thus, p(m + 1) is true.
By the principle of mathematical induction, p(n) is true for all $\text{n}\in\text{N}.$
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