Let P n denote the statement that n 2- n is odd- Then-A- n-1B- nC- None of theseD- n-2

The correct option is D n gt; 2 P(1): 1^2+1 is odd→ not true P(2): 2^2+2 is odd→ not true Suppose P(k) is true. k^2+k is odd⇒ k^2+k=2m+1 P(k+1): (k+1) ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XI

Mathematics

Proof by mathematical induction

Let P n denot...

Question

Let P(n) denote the statement that $n^{2}$ + n is odd. Then,

n > 1

No worries! We‘ve got your back. Try BYJU‘S free classes today!

n > 2

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

None of these

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is C
n > 2

$P (1) : 1^{2} + 1 is odd \to n o t t r u e P (2) : 2^{2} + 2 is odd \to n o t t r u e$

Suppose P(k) is true.
$k^{2} + k is odd \Rightarrow k^{2} + k = 2 m + 1 P (k + 1) : (k + 1)^{2} + (k + 1) = (k^{2} + 2 k + 1) + (k + 1) = (k^{2} + k) + (2 k + 2) = 2 m + 1 + 2 k + 2 2 (m + k + 1) + 1 \Rightarrow P(k+1) is true$

Suggest Corrections

Similar questions

Let P(n) denote the statement that $n^{2}$ + n is odd. It

is seem that P(n) ⇒ P(n + 1), $P_{n}$ is true for all

Let P(n) denote the statement that $n^{2}$ + n is odd. It

is seem that P(n) ⇒ P(n + 1), $P_{n}$ is true for all

Let $f (n) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} n & n + 1 & n + 2^{n} P_{n} & ^{n + 1} P_{n + 1} & ^{n + 1} P_{n + 2}^{n} C_{n} & ^{n + 1} C_{n + 1} & ^{n + 1} C_{n + 2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
Then, f(n) is divisible by

Q. The statement P(n) =

4^{n} - 1 \geq 3^{n}

is true for which of the following options?

Q. Let

P (n)

be the statement that

n^{2} - n + 41

is prime, then which of the following is not true ?

Join BYJU'S Learning Program

Let P(n) denote the statement that n2 + n is odd. Then,

The correct option is C n > 2 P(1):12+1 is odd→not trueP(2):22+2 is odd→not true Suppose P(k) is true. k2+k is odd⇒k2+k=2m+1P(k+1):(k+1)2+(k+1)=(k2+2k+1)+(k+1)=(k2+k)+(2k+2)=2m+1+2k+22(m+k+1)+1⇒P(k+1) is true

Let P(n) denote the statement that $n^{2}$ + n is odd. Then,