(1)/((a+1)(a+2))-(1)/((a+2)(a+3))+(1)/((a+1)(a+3))

Question

how is the answer: (a+4)/((a+1)(a+2)(a+3))

Answer 1

Question: Simplify (1)/((a+1)(a+2))-(1)/((a+2)(a+3))+(1)/((a+1)(a+3)).

Our expression is: E = T1 - T2 + T3, where

T1 = 1/(a+1)(a+2) = (a+3)/(a+1)(a+2)(a+3)

T2 = 1/(a+2)(a+3) = (a+1)/(a+1)(a+2)(a+3)

T3 = 1/(a+1)(a+3) = (a+2)/(a+1)(a+2)(a+3)

Now that the three terms all have the same denominator, we can simply add together the numerators

E = N/D

where D = (a+1)(a+2)(a+3)

and N = (a+3) - (a+1) + (a+2) = a + 4

Answer: E = N/D = (a+4)/(a+1)(a+2)(a+3)