Let S(K)=1+3+5+ +(2 K-1)=3+K^2 . Then which of the following is true?

Correct answer: D. S(K) S(K+1) aligned & S(k)=1+3+5+ . .+(2 k-1)=3+k² & S(k+1)=1+3+5+ +(2 k-1)+(2 k+1) & =(3+k²)+2 k+1=k²+2 k+4[ from S(k)=3+k²] & =3+(k^2+2 k+1)=3+(k+1)^2=S(k+1) aligned Although S(k) in itself is not true but it considered true will always imply towards S(k+1) .

JEE Main 2004

Exam: JEE Main
Year: 2004
Subject: Mathematics
Chapter: Mathematical Induction

JEE Main 2004 · Mathematics · Mathematical Induction

Let S(K)=1+3+5+ +(2 K-1)=3+K^2 . Then which of the following is...

Let

S(K)=1+3+5+\ldots+(2 K-1)=3+K^2

. Then which of the following is true?

Options

A $S(1)$ is correct
BPrinciple of mathematical induction can be used to prove the formula
C $S(K) \neq S(K+1)$
D $\mathrm{S}(\mathrm{K}) \Rightarrow \mathrm{S}(\mathrm{K}+1)$

Correct answer

D. $\mathrm{S}(\mathrm{K}) \Rightarrow \mathrm{S}(\mathrm{K}+1)$

Step-by-step solution

\begin{aligned}& S(k)=1+3+5+\ldots \ldots . .+(2 k-1)=3+k² \\ & S(k+1)=1+3+5+\ldots \ldots \ldots+(2 k-1)+(2 k+1) \\ & =\left(3+k²\right)+2 k+1=k²+2 k+4\left[\text{ from }S(k)=3+k²\right] \\ & =3+\left(k^2+2 k+1\right)=3+(k+1)^2=S(k+1) \end{aligned}

Although

S(k)

in itself is not true but it considered true will always imply towards

S(k+1)

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