First slide

Introduction to P.M.I

Question

Let $S (K) : 1 + 3 + 5 + \dots + (2 K - 1) = 3 +$ $k^{2}$
Then which of the following is true?

Moderate

A

$S (K) \neq S (K + 1)$
B

$S (K) \Rightarrow S (K + 1)$
C

$S (1)$ is correct
D

principle of mathematical induction can be used to prove the formula

Solution

$S (1) : 1 = 3 + 1^{2}$ which is not true. Suppose
$S (K)$ is true, then
$1 + 3 + 5 + \dots + (2 K - 1) = 3 + K^{2}$
Adding $(2 K + 1)$ to both the sides, we get
$\begin{array}{l} 1 + 3 + 5 + \dots + (2 K - 1) + (2 K + 1) = 3 + K^{2} + 2 K + 1 \\ = 3 + (K + 1)^{2} \end{array}$
which is $S (K + 1)$
Thus, $S (K) f i S (K + 1) .$

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