Question

Let $T(k)$ be the statement $1 + 3 + 5 + ... + (2k - 1)= k^2 +10$ Which of the following is correct?

• A. T(1) is true
• B. T(k) is true $\Rightarrow$ T(k + 1) is true
• C. T(n) is true for all n $\in$ N
• D. All above are correct

Answer 1

Answer: (B)

T(k) is true $\Rightarrow$ T(k + 1) is true

Answer 2

When $k = 1$, LHS $= 1$ but RHS $= 1 + 10 = 11$ $\therefore T(1)$ is not true Let $T(k)$ is true. That is $1+ 3+ 5+.....+ (2k -1) = k^2 +10$ Now, $1+3+5+.....+(2k -1)+(2k +1)$ $= k^2 +10 + 2k +1 = (k +1)^2 +10$ $\therefore T(k +1)$ is true. That is $T(k)$ is true $\Rightarrow T(k +1)$ is true. But $T(n)$ is not true for all $n \in N$ , as $T(1)$ is not true.