Problem Solving

The sequence of numbers a1,a2,a3.....an is defined by an = 1/n - 1/n+2 for each integer n >or= 1. What is the sum of the first 20 terms of this sequence?

A) (1+1/2) - 1/20
B) (1+1/2) - (1/21+1/22)
C) 1 - (1/20 + 1/20)
D) 1 - 1/22
E) 1/20 - 1/22

Hi,
t1 = 1... -1/3
t2 = ..1/2.....-1/4
t3 = ......1/3......-1/5
t4 = ............1/4......-1/6
.
.
.
t18 = ...................................... 1/18....-1/20
t19 = ...........................................1/19......-1/21
t20 = .................................................1/20.......-1/22
------------------------------------------------------------------------
Sum = 1+1/2+0+0+0...................................+0..-1/21 -1/22
So, sum = (1+1/2) - (1/21+1/22)

Hence, B

Thanks Frankenstein. Should you be doing the calculations or just cancelling out the fractions with each other? I think you have to lay them out all first then cancel. But this will take much time...any faster approach?

MBA.Aspirant wrote:Thanks Frankenstein. Should you be doing the calculations or just cancelling out the fractions with each other? I think you have to lay them out all first then cancel. But this will take much time...any faster approach?

Hi,
Just cancelling. No calculations. If you want faster approach, we can perhaps write in a different manner even though both are same.
write all values of 1/n as 1,1/2,...1/20
then write all values of 1(n+2) as 1/3,1/4,...1,22
While subtracting, we can see that common elements from 1/3 to 1/20 are cancelled and we are left with 1+1/2 from first series and (1/21+1/22) from the second series.