Q.

The identity 13+23+33+…+n3 is equal to

see full answer

Talk to JEE/NEET 2025 Toppers - Learn What Actually Works!

Real Strategies. Real People. Real Success Stories - Just 1 call away
An Intiative by Sri Chaitanya

a

n(n−1)22

b

n(n+1)2

c

{n(n+1)}22

d

n(n+1)22

answer is D.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

Let the given statement be P(n).P(n):13+23+33+…+n3=n(n+1)22Step I : For n=1,             P(1):1(1+1)22=1×222=12=1=13 which is trueStep ll: Let it is true for n = k,             13+23+33+…+k3=k(k+1)22------iStep lll: For n=k+1,             13+23+33+43+…+k3+(k+1)3           =k(k+1)22+(k+1)3      [using Eq. (i)]            =k2(k+1)24+(k+1)31=k2(k+1)2+4(k+1)34  On taking (k + 1)2 common in numerator Part,            =(k+1)2k2+4(k+1)4=(k+1)2k2+4k+44=(k+1)2(k+2)24=(k+1)2[(k+1)+1]24=(k+1){(k+1)+1}22Therefore, P(k + 1) is true when P(k) is true. Hence, from the principle of mathematical induction, the statement is true for all natural numbers n.
Watch 3-min video & get full concept clarity
score_test_img

Get Expert Academic Guidance – Connect with a Counselor Today!

whats app icon