Question

Question ▶

1.) 1×2+2×3+3×4--------- +n(n+1)=
[n(n+1)(n+2)/3]


2.) For all
$n \geqslant 1 \: prove \: that \:$

Answer 1

Show Truth for N = 1

LHS = (1) (2) = 2

RHS =

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

Which is Equal to 2

Assume N = K

(1)(2)+(2)(3)+(3)(4)+⋯+(k)(k+1)=(k)(k+1)(k+2)/3(1)(2)+(2)(3)+(3)(4)+⋯+(k)(k+1)=(k)(k+1)(k+2)/3

Proof that the equation is true for N = K + 1

(1)(2)+(2)(3)+(3)(4)+⋯+(k)(k+1)+(k+1)(k+2)(1)(2)+(2)(3)+(3)(4)+⋯+(k)(k+1)+(k+1)(k+2)

Which is Equal To:

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

This is where I've went so far

If I did the calculation right the Answer should be

(k+1)(k+2)(k+3)/3
Your proof is fine, but you should show clearly how you got to the last expression.

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

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

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

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

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

You should also word your proof clearly. For example, you can say "Let P(n)P(n) be the statement ... P(1)P(1) is true ... Assume P(k)P(k) is true for some positive integer kk... then P(k+1)P(k+1) is true ... hence P(n)P(n)is true for all positive integers nn"

Answer 2

Heyy mate ❤✌✌❤

Here's your Answer