JEE Main2004MathematicsMathematical InductionActual
Let S(K)=1+3+5+ +(2 K-1)=3+K^2 . Then which of the following is true?
Options
- AS(1) is correct
- BPrinciple of mathematical induction can be used to prove the formula
- CS(K) S(K+1)
- DS ( K ) S ( K +1)
Correct answer
D. S ( K ) S ( K +1)
Step-by-step solution
aligned & S(k)=1+3+5+ . .+(2 k-1)=3+k^2 & S(k+1)=1+3+5+ +(2 k-1)+(2 k+1) & = (3+k^2 )+2 k+1=k^2+2 k+4 [ from S(k)=3+k^2 ] & =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) .