First slide

Introduction to P.M.I

Question

Using mathematical induction, the numbers a_ns are defined by $a_{0} = 1, a_{n + 1} = 3 n^{2} + n + a_{n}, (n \geq 0)$ Then, a_n is equal to

Moderate

A

$n^{3} + n^{2} + 1$
B

$n^{3} - n^{2} + 1$
C

$n^{3} - n^{2}$
D

$n^{3} + n^{2}$

Solution

$\begin{array}{l} Given a_{0} = 1, a_{n + 1} = 3 n^{2} + n + a_{n} \\ \Rightarrow a_{1} = 3 (0) + 0 + a_{0} = 1 \\ \Rightarrow a_{2} = 3 (1)^{2} + 1 + a_{1} = 3 + 1 + 1 = 5 \end{array}$
From option (b)
$\begin{array}{l} Let P (n) = n^{3} - n^{2} + 1 \\ ∴ P (0) = 0 - 0 + 1 = 1 = a_{0}, P (1) = 1^{3} - 1^{2} + 1 = 1 = a_{1} \\ and P (2) = (2)^{3} - (2)^{2} + 1 = 5 = a_{2} \end{array}$

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