1/6 n (n+1) ( n+2) - 1/6 n-1 (n) (n+1) = 1/2 n (n+1)

I am solving this with the assumption that all the n terms are in the numerator and 6 is in the denominator. (That's the only way to arrive at the answer.)
The factors common in both sides can be taken out.

(1/6)n(n+1) can be taken out

1/6 n (n+1) ( n+2) - 1/6 n-1 (n) (n+1)

= 1/6 n (n+1) [(n+2)-(n-1)]
=1/6 n (n+1) [n + 2 - n + 1)]
=1/6 n (n+1) [n - n + 2 + 1]
=1/6 n (n+1) [2 + 1]
=1/6 n (n+1) [3]
= 3/6 n (n+1)
= 1/2 n (n+1)

marmar29 wrote:hello everyone

i really do stuck in this equaiton to make it simple one |& get the final results
1/6 n (n+1) ( n+2) - 1/6 n-1 (n) (n+1) = 1/2 n (n+1)
please show me the steps
thanks

First up, what's the question? Solve for a possible value of n? Solve for all the values of n? Without the question (and choices), it's impossible to find the best possible approach.

Assuming that we need all the values of n, I'd start with Brett's analysis:

n is common to all 3 terms, so n=0 works (you'd get 0 - 0 = 0, which is correct). Eliminate any choices that don't contain 0.

(n+1) is common to all 3 terms, so n=-1 works (you'd again get 0 - 0 = 0, which is correct). Eliminate any remaining choices that don't contain -1.

Assuming that there's more than 1 choice remaining, divide both sides by (n)(n+1) to get:

(1/6)(n+2) - (1/6)(n-1) = 1/2

(Note: we can only do this step because we're now assuming that n doesn't equal 0 or -1.)

Multiply both sides by 6:

(n+2) - (n-1) = 3

Simplify:

3 = 3

No more "n"s, so there are no other possible values.