Question

Use induction to prove the following: (b)1.2 + 2.3 + 3.4 +...+(n-1)n= (n^3-n)/3

Answer 1

Answer:

We shall prove the result by principle of mathematical induction.  

checking for n = 1,  

LHS : 1.2 = 2

RHS : 1/3 * 1 * 2 * 3 = 2.

Hence true for n = 1

Let us assume the result is true for n = k.  

that is, 1.2 + 2.3 +.....k(k+1) = 1/3 * k * (k+1) * (k+2)  

We shall prove the result to be true for n = k+1.  

that is, to prove 1.2 + 2.3 .....+ k(k+1) + (k+1)(k+2) = 1/3 (k+1) (k+2) (k+3)

consider LHS:

1.2 + 2.3 .....+ k(k+1) + (k+1)(k+2)  

= 1/3 * k * (k+1) * (k+2) + (k+1)(k+2)

=(k+1)(k+2) [1/3*k + 1]

=(k+1)(k+2)(k+3)1/3

=RHS.  

Hence the result holds for n=k+1.  

Hence proof is complete by PMI and therefore the result holds.  

Step-by-step explanation: