If P(n): " 2^{2n}-1 is divisible by k for all | vedclass

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(B) Given the statement $P(n):$ " $2^{2n}-1$ is divisible by $k$ for all $n \in N$ ".We can write $2^{2n}-1$ as $(2^2)^n - 1 = 4^n - 1$.Using the bino

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$3$

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(B) Given the statement $P(n):$ " $2^{2n}-1$ is divisible by $k$ for all $n \in N$ ".We can write $2^{2n}-1$ as $(2^2)^n - 1 = 4^n - 1$.Using the binomial expansion,$4^n - 1 = (3+1)^n - 1$.Expanding $(3+1)^n$ using the binomial theorem: $(3+1)^n = 1 + n(3) + \frac{n(n-1)}{2}(3^2) + \dots + 3^n$.So,$4^n - 1 = (1 + 3n + 9\frac{n(n-1)}{2} + \dots + 3^n) - 1 = 3n + 9\frac{n(n-1)}{2} + \dots + 3^n$.This expression is clearly divisible by $3$ for all $n \in N$.Thus,the value of $k$ is $3$.

If $P(n):$ " $2^{2n}-1$ is divisible by $k$ for all $n \in N$ " is true,then the value of $k$ is:

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