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Principle of Mathematical Induction

Short Answer Type

11.

If A = $[\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}]$ , then by the principle of Mathematical induction, prove that Aⁿ = $[\begin{matrix} 1 & 2 n \\ 0 & 1 \end{matrix}]$

Multiple Choice Questions

12.

The value of 2, 6, 10 ... (4n - 6)(4n - 2) is equal to

C(2n, n)
(n + 1)(n + 2)(n + 3) ... (2n)
n! P (2n, n)
None of above

13.
$Using mathematical induction, the numbers a_{n}' s are defined by, a_{0} = 1, a_{n + 1} = 3 n^{2} + n + a_{n}, (n \geq 0) Then a_{n} = ?$

n³ + n² + 1

n³ - n² + 1

n³ - n²

n³ + n²

n³ - n² + 1

$Given, a_{0} = 1, a_{n + 1} = 3 n^{2} + n + a_{n} \Rightarrow a_{1} = 3 (0) + 0 + a_{0} = 1 and a_{2} = 3 {(1)}^{2} + 1 + a_{1} = 3 + 1 + 1 = 5 From option (b), Let P (n) = n^{3} - n^{2} + 1 ∴ P (0) = 0 - 0 + 1 = 1 = a_{0} P (1) = 1^{2} - 1^{2} + 1 = 1 = a_{1} P (2) = 2^{2} - 2^{2} + 1 = 1 = a_{2} Hence option (b) is correct$