Use Induction to prove that1x2 2x3 3x4 nn1 nn1n23 for al

Use Induction to prove that1x2 + 2x3 + 3x4 +.....+ n(n+1) = n(n+1)(n+2)/3 for all n.

Solution

Consider n(n+1)(n+2) - (n-1)n(n+1) = n(n+1)[n+2-n+1] = 3n(n+1) So we now consider a series of equations as follows: n(n+1)(n+2) - (n-1)n(n+1) = 3n(n+1) (n-1)n(n+1) - (n-2)(n-1)(n) = 3(n-1)n (n-2)(n-1)n - (n-3)(n-2)(n-1) = 3(n-2)(n-1) ----------------------------- ----------- ----------------------------- ----------- 3 x 4 x 5 - 2 x 3 x 4 = 3(3)(4) 2 x 3 x 4 - 1 x 2 x 3 = 3(2)(3) Add all the 1 x 2 x 3 - 0 x 1 x 2 = 3(1)(2) equations ----------------------------------------------- n(n+1)(n+2) - 0 = 3[1x2 + 2x3 + ... + n(n+1)] Note that when adding all the equations there is a cancelling of terms on the lefthand side between adjacent lines. The term -(n-1)n(n+1) in the first line cancels with the term +(n-1)n(n+1) in the second line, and so on down all the equations. The only terms that do not cancel are the first term in the top equation and the second term in the last equation. n(n+1)(n+2) We then get ----------- = 1x2 + 2x3 + 3x4 + .... + n(n+1) 3