Consider the statement $P(n): n^2 - n + 37$ is prime. Which one of the following is true?
- A$P(3)$ is false,but $P(5)$ is true.
- B$P(5)$ is false,but $P(3)$ is true.
- CBoth $P(3)$ and $P(5)$ are true.
- DBoth $P(3)$ and $P(5)$ are false.
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Use the Principle of Mathematical Induction to prove that $\cos \theta \cos 2 \theta \cos 2^{2} \theta \ldots \cos 2^{n-1} \theta = \frac{\sin 2^{n} \theta}{2^{n} \sin \theta}$ for all $n \in N$.
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View SolutionUsing mathematical induction,the numbers $a_n$ are defined by:
$a_0 = 1, a_{n+1} = 3n^2 + n + a_n, (n \geq 0)$.
Then,$a_n$ is equal to:
$a_0 = 1, a_{n+1} = 3n^2 + n + a_n, (n \geq 0)$.
Then,$a_n$ is equal to:
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View SolutionLet $P(n) : 3^n < n!$ for $n \in N$ be true for $n \geq \lambda$. Find the smallest value of $\lambda$.
Prove that for all $n \in N$,$41^{n}-14^{n}$ is a multiple of $27$ using the principle of mathematical induction.
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View SolutionProve the following by using the principle of mathematical induction for all $n \in N$:
$a + ar + ar^{2} + \ldots + ar^{n-1} = \frac{a(r^{n} - 1)}{r - 1}$
$a + ar + ar^{2} + \ldots + ar^{n-1} = \frac{a(r^{n} - 1)}{r - 1}$
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