Home
Class 12
MATHS
Prove by mathematical induction that 1^3...

Prove by mathematical induction that `1^3+2^3+……+n^3=[(n(n+1))/2]^2`

Text Solution

Verified by Experts

Let `P(n):1^3+2^3+3^3+.....+n^3=[(n(n+1))/(2)]^2`......(i)
Step I For `n=1`, LHS of Eq.(i) `=^3=1` and RHS of Eq. (i). `[(1(1+1))/(2)]^2=1^2=1`
`therefore LHS=RHS`
Therefore ,P(1), is ture.
Step II Assume P(k) is true , then
`P(k):1^3+2^3+3^3+.....K^3=[(k(k+1))/(2)]^2`
Step III For `n=k+1`,
`P(k+1):1^3+2^3+3^3+......+^3(k+1)^3`
`=[((k+1)+(k+2))/(2)] ^2`
LHS `=1^3+2^3+3^3+.....+k^3+(k+1)^3=[(k(k+1))/(2)]^2+(k+1)^3` [by assumption step]
`=((k+1)^2)/(4)[k^2+4(k+1)]`
`=((k+1)^2(k^2+4k+4))/(4)`
`=((k+1)^2(k+2)^2)/(4)`
`=[(k+1(k+2))/(2)]^2=RHS`
Therefore , `P(k+1)` is true , Hence , by the principle of mathematical induction , P(n)is true for all `n epsi N`.
Promotional Banner

Similar Questions

Explore conceptually related problems

Prove by mathematical induction 1+2+3+……+n(n(n+1))/(2) .

Prove by mathematical induction that sum_(r=0)^(n)r^(n)C_(r)=n.2^(n-1), forall n in N .

Prove by mathematical induction that 10^(2n-1)+1 is divisible by 11

if a,b,c,d,e and f are six real numbers such that a+b+c=d+e+f a^2+b^2+c^2=d^2+e^2+f^2 and a^3+b^3+c^3=d^3+e^3+f^3 , prove by mathematical induction that a^n+b^n+c^n=d^n+e^n+f^n forall n in N .

Prove the following by the principle of mathematical induction: 1^2+2^2+3^2++n^2=(n(n+1)(2n+1))/6

Prove by mathematical induction that 1.4+4.7+7.10+…. up to n terms =n(3n^(2)+3n-2)

If a_(1)=1, a_(2)=5 and a_(n+2)=5a_(n+1)-6a_(n), n ge 1 , show by using mathematical induction that a_(n)=3^(n)-2^(n)

if a+b=c+d and a^2+b^2=c^2+d^2 , then show by mathematical induction a^n+b^n=c^n+d^n

Use the principle of mathematical induction to show that 5^(2n+1)+3^(n+2).2^(n-1) divisible by 19 for all natural numbers n.

Let u_(1)=1,u_2=2,u_(3)=(7)/(2)and u_(n+3)=3u_(n+2)-((3)/(2))u_(n+1)-u_(n) . Use the principle of mathematical induction to show that u_(n)=(1)/(3)[2^(n)+((1+sqrt(3))/(2))^n+((1-sqrt(3))/(2))^n]forall n ge 1 .