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

Principle of mathematical Induction can be used to prove the formula
B

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

$S (k) \Rightarrow S (k + 1)$
D

$S (1)$ is correct

Solution

Let $S (k)$ be true. Then
$1 + 3 + 5 + \dots + (2 k - 1) = 3 + k^{2} . . . . (i)$
Now,
$S (k + 1) = 1 + 3 + 5 + \dots + (2 k - 1) + (2 k + 1)$
$= (3 + k^{2}) + 2 k + 1 = 3 + (k + 1)^{2}$ [Using (i)]
$\Rightarrow S (k + 1)$ is true.
Hence, $S (k) \Rightarrow S (k + 1)$ .

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