Prove ∑r^3 = 1/4 n^2(n+1)^2

Proof by induction

Base Case when n=1 

LHS = 1^3=1 RHS= 1/4(1^2)(1+1)^2=1/4(1)(2^2)=1/4(4)

Assume true for n=k 

so  ∑r^3= 1/4k^2(k+1)^2

For n=k+1 

∑r^3 = ∑k terms + (k+1)^3 = 1/4(k^2)(k+1)^2 + (k+1)^3

= 1/4(k^2)(k+1)^2 + (k+1)^2(k+1)

=1/4(k^2+4k+4)(k+1)^2 

Completing the square k^2+ 4k + 4 = (k+2)^2

=1/4(k+2)^2(k+1)^2

Same form as above with n replaced by k+1

Therefore it is true for n=k+1 if true for n=k but true for n=1 so true for n=2 and so on.

Related Further Mathematics A Level answers

All answers ▸

How to integrate ln(x)?


Given that x = i is a solution of 2x^3 + 3x^2 = -2x + -3, find all the possible solutions


Find the eigenvalues and corresponding eigenvectors of the following matrix: A = [[6, -3], [4, -1]]. Hence represent the matrix in diagonal form.


Find the eigenvalues and eigenvectors of A = ([2, 0 , 0], [0, 1, 1], [0, 3, 3])


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences