Question

Prove by using principle of mathematical induction: $4^{3}+8^{3}+12^{3}+..... up to n terms =16n^{2}(n+1)^{2}$

Answer 1

Dear Student,  Please find below the solution to the asked query:  43+83+123...upto n terms =16n2(n+1)243+83+12+....+(4n)3=16n2(n+1)2For n=1P(1): 43=16(1)2(1+1)2⇒64=64  so its trueAssume that P(k) is true  for some positive integer k : we have 43+83+123+.....+(4k)2=16k2(k+1)2Now we shall prove that P(k+1) is also true So 43+83+123+....+(4k)3+43(k+1)3⇒16k2(k+1)2+43(k+1)3⇒16(k+1)2[k2+4(k+1)]⇒16(k+1)2[k2+4k+4]⇒16(k+1)2(k+2)2⇒16(k+1)2[(k+1)+1]So hence its true