Clearly nothing is going to cancel in the usual straight forward way, so I tried rewriting the numbers. As I observed in my previous post, one can write the terms you circled in terms of the other terms in the series, that is ( sorry, too useless to use latex !)

1/20 = 1/5- 3/20

1/28 = 1/7- 3/28

1/36 = 1/9 - 3/36...

Substituting the above into the series, gives the following:

3/4 - 1/3 + (1/5- 3/20) + 3/12 - 1/5 + (1/7- 3/28) + 3/20 - 1/7 + (1/9 - 3/36) +....

Now the terms cancel ! And we are left with the final terms in "n" as given in the markscheme and 3/4 - 1.3 + 3/12 = 3/4 -1/3 + 1/4.
Tricky question for an A-level though!

(edited 5 years ago)

Reply 8

JacobPhysics

OP

14

Original post by bluenotewitt

Maybe I can help you after all... :smile:

Clearly nothing is going to cancel in the usual straight forward way, so I tried rewriting the numbers. As I observed in my previous post, one can write the terms you circled in terms of the other terms in the series, that is ( sorry, too useless to use latex !)

1/20 = 1/5- 3/20

1/28 = 1/7- 3/28

1/36 = 1/9 - 3/36...

Substituting the above into the series, gives the following:

3/4 - 1/3 + (1/5- 3/20) + 3/12 - 1/5 + (1/7- 3/28) + 3/20 - 1/7 + (1/9 - 3/36) +....

Now the terms cancel ! And we are left with the final terms in "n" as given in the markscheme and 3/4 - 1.3 + 3/12 = 3/4 -1/3 + 1/4.
Tricky question for an A-level though!

Ah yeah that’s the trickiest question I’ve seen on this topic so far... thanks for your help :smile:

Reply 9

DFranklin

18

Original post by JacobPowell

Ah yeah that’s the trickiest question I’ve seen on this topic so far... thanks for your help :smile:

The thing to do is look at the partial fraction decomposition: you have a 3/4 term, a 1/4 term and a -1 term,
(

\frac{3}{4}\frac{1}{2r-1}, \frac{1}{4} \frac{1}{2r+3}, -1 \frac{1}{2r+1}

), which gives you a clue about how the terms are going to cancel.

It would also probably help to keep the 3/4, 1/4 fractions in when writing out (rather than multiplying 1/4 x 1/5 = 1/20 for example):