P n - 1-3-5-2 n -1- n 2The statement Pn isA- is true for all natural numbersB- is not true for n-1C- none of theseD- is true for n-2

The correct option is A is true for all natural numbers P(n):1+3+5+...+2n 1=n^2 P(1): 1=1^2 ⇒ 1=1 true Assume P(k) is true 1+3+5+...+2k 1=k^2 Now, P(k+1 ...

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

Byju's Answer

Standard XI

Mathematics

Proof by mathematical induction

P n : 1+3+5+…...

Question

$P (n) : 1 + 3 + 5 + . . . + 2 n - 1 = n^{2}$

The statement P(n) is

is true for all natural numbers

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

is not true for n>1

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

is true for n>2

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

none of these

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

Open in App

Solution

The correct option is A
is true for all natural numbers

$P (n) : 1 + 3 + 5 + . . . + 2 n - 1 = n^{2} P (1) : 1 = 1^{2} \Rightarrow 1 = 1 - t r u e$
Assume P(k) is true
$1 + 3 + 5 + . . . + 2 k - 1 = k^{2} N o w, P (k + 1) : 1 + 3 + 5 + . . . + 2 n - 1 + 2 k + 1 = k^{2} + 2 k + 1 = (k + 1)^{2}$

P(k+1) is also true.
Hence, P(n) is true for all natural numbers

Suggest Corrections

Similar questions

$P (n) : 1.2 + 2.3 + 3.4 + . . . + n (n + 1) = \frac{(n + 1) (n + 2)}{3}$

The statement P(n) is

$P (n) : 1 + 3 + 3^{2} + . . . + 3^{n - 1} = \frac{3^{n} - 1}{2}$
The statement P(n)

Q. Let

P (n)

be the statement

2^{n} < n!

where

n

is a natural number, then

P (n)

is true for:

'For all natural numbers N, if P(n) is a statement about n and P(k+1) is true if P(k) is true for an arbitrary natural number k, then P(n) is always true.' State true or false.

Let P(n) be the statement : $2^{n} \leq 3 n$ . If P(r) is true, show that P(r + 1) is true. Do you conclude that P(n) is true for all $n ϵ N$ .

Join BYJU'S Learning Program

P(n):1+3+5+...+2n−1=n2 The statement P(n) is

The correct option is A is true for all natural numbers P(n):1+3+5+...+2n−1=n2P(1):1=12⇒1=1− true Assume P(k) is true 1+3+5+...+2k−1=k2Now, P(k+1):1+3+5+...+2n−1+2k+1=k2+2k+1=(k+1)2 P(k+1) is also true. Hence, P(n) is true for all natural numbers

$P (n) : 1 + 3 + 5 + . . . + 2 n - 1 = n^{2}$

The statement P(n) is