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Let \(P(n): 2+2^2+2^3+\ldots+2^n=2^{n+1}, n \in \mathbf{N}\). Then,
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Verified Answer
The correct answer is:
\(P(m)\) is true \(\Rightarrow P(m+1)\) is true
Given,
\(p(n)=2+2^2+2^3+\ldots .+2^n=2^{n+1}\)
Where \(n \in \mathrm{N}\).
Let \(p(m)\) is true then,
\(P(m)=2+2^2+2^3+\ldots .+2^m=2^{m+1}\)
So,
\(\begin{aligned}
P(m+1) & =2+2^2+2^3+\ldots .+2^m+2^{m+1} \\
& =\left(2+2^2+2^3+\ldots .+2^m\right)+2^{m+1} \\
& =2^{(m+1)}+2^{m+1}=2.2^{m+1}=2^{m+2}
\end{aligned}\)
Clearly, if \(P(m)\) is true then \(p(m+1)\) is also true.
\(\therefore P(m)\) is true \(\Rightarrow P(m+1)\) is true.
\(p(n)=2+2^2+2^3+\ldots .+2^n=2^{n+1}\)
Where \(n \in \mathrm{N}\).
Let \(p(m)\) is true then,
\(P(m)=2+2^2+2^3+\ldots .+2^m=2^{m+1}\)
So,
\(\begin{aligned}
P(m+1) & =2+2^2+2^3+\ldots .+2^m+2^{m+1} \\
& =\left(2+2^2+2^3+\ldots .+2^m\right)+2^{m+1} \\
& =2^{(m+1)}+2^{m+1}=2.2^{m+1}=2^{m+2}
\end{aligned}\)
Clearly, if \(P(m)\) is true then \(p(m+1)\) is also true.
\(\therefore P(m)\) is true \(\Rightarrow P(m+1)\) is true.
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