Porisma

Worked solution

Subtract the correct prefix from a finite sum

Given ∑r=0n1/[(r+1)(r+2)(r+3)]=14−1/[2(n+2)(n+3)]\sum_{r=0}^n1/[(r+1)(r+2)(r+3)]=\tfrac14-1/[2(n+2)(n+3)], deduce ∑r=5201/[(r+1)(r+2)(r+3)]\sum_{r=5}^{20}1/[(r+1)(r+2)(r+3)]. marks

[2010/MI/Promo/5(modified)]

CJC · Review problems · Question 4

Answer

58/531358/5313.

Remove exactly the terms before r=5

Write F(n)=1/4−1/[2(n+2)(n+3)]F(n)=1/4-1/[2(n+2)(n+3)]. The required sum is F(20)−F(4)F(20)-F(4), not F(20)−F(5)F(20)-F(5).

How did this part go?

Evaluate the remaining two boundary terms

F(20)−F(4)=−1/[2(22)(23)]+1/[2(6)(7)]=58/5313F(20)-F(4)=-1/[2(22)(23)]+1/[2(6)(7)]=58/5313.

How did this part go?

Common mistakes

Subtracting the prefix through 5 would discard one required term.

Read the hint