Question

Use the mathematical induction to prove that

1.2.3 + 2.3.4 + ⋯ + ( + 1)( + 2) =

( + 1)( + 2)( + 3)

4

Where n is positive integer​

Answer 1

Answer:

Step-by-step explanation:

We shall prove the result by principle of mathematical induction.  

checking for n=1,  

LHS:1.2=2

RHS:  

3

1 ×1×2×3=2.

Hence true for n=1

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

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

3 1 ×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)=  

3 1 (k+1)(k+2)(k+3)

consider LHS:1.2+2.3.....+k(k+1)+(k+1)(k+2)

= 31 ×k×(k+1)×(k+2)+(k+1)(k+2)

=(k+1)(k+2)[  

(k+1)]

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

31=RHS.  

Hence the result holds for n=k+1.

Answer 2

Answer:

Step-by-step explanation:

To prove that 1.2.3 + 2.3.4 + ... + n(n+1)(n+2)/4 = (n+1)(n+2)(n+3)/4 using mathematical induction, we need to show that the formula holds for the base case (n = 1), and then assume that the formula holds for some arbitrary positive integer k, and prove that it also holds for k+1.