# Let P(n) be the statement `n^{2} + n is even’. If P(k) is true , then show that P(k + 1) is true.

### Asked by Topperlearning User | 4th Jun, 2014, 01:23: PM

Expert Answer:

### We are given that P(k) is true.
i.e. `k^{2} + k is even’
Let k^{2} + k = 2q, q _{}N
Let n = k + 1
_{} (k + 1)^{2} + k + 1 = (k + 1)(k + 1) + k +1 = k(k + 1) + 2(k + 1)
= 2q + 2(k + 1), which is even.
Thus P(k +1) is true, whenever P(k) is true.

^{2}+ k is even’

^{2}+ k = 2q, q

_{}N

_{}(k + 1)

^{2}+ k + 1 = (k + 1)(k + 1) + k +1 = k(k + 1) + 2(k + 1)

### Answered by | 4th Jun, 2014, 03:23: PM

## Concept Videos

- prove by mathematical induction,cos
- By using principle of mathematical induction, prove that: 1 + 4 + 7 + --- + (3n - 2) =n(3n-1)
- Let P(n) be the statement `n(n + 1) is an even number’ then find P(6).
- If P(n) be the statement `12n + 3’ is a multiple of 5, then show that P(3) is false, whereas P(6) is true.
- Let P(n) be the statement . If P(k) is true,then show that P(k+1) is true.
- Explain the principle of mathematical induction:
- If P(n) be the statement ‘n
^{2}– n + 41 is a prime number’, then show that P(1), P(2) are true but P(41) is not true. - Show by using principle of mathematical induction, prove that: 1 + 2 + 3 + -----+ n =
- Let P(n) be the statement , If P (k) is true , then show that P(k + 1) is true.

### Kindly Sign up for a personalised experience

- Ask Study Doubts
- Sample Papers
- Past Year Papers
- Textbook Solutions

#### Sign Up

#### Verify mobile number

Enter the OTP sent to your number

Change