Eric Nicholas K's answer to LockB's Secondary 3 A Maths Singapore question.
done
{{ upvoteCount }} Upvotes
clear
{{ downvoteCount * -1 }} Downvotes
This method of cancellations is known as the method of differences. The sum is made simple because the majority of the terms present in the full series actually cancel out, leaving behind a tiny minority of surviving terms to be calculated easily.
Date Posted:
3 years ago
i dont really understand the first 4 columns of the expansion in part b. also, why is it minus but not plus or times tho
The fraction we have decomposed into is 1 / (x + 1)(x + 2) = 1 / (x + 1) - 1 / (x + 2).
When x = 1, we have
1 / (2) (3) = 1/2 - 1/3
When x = 2, we have
1 / (3) (4) = 1/3 - 1/4
The rest is similar.
The sum of all these goes like this.
1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + 1/5 - 1/6 + 1/6 - 1/7 + 1/7 - 1/8 + ... + ... + 1/197 - 1/198 + 1/198 - 1/19 9 + 1/199 - 1/200
and you will see that there are many pairs cancelling out (1/3 against 1/3, 1/4 against 1/4 and so on, up to 1/199 against 1/199).
Surviving are the 1/2 and the -1/200, so combining these two, we get sum = 1/2 - 1/200 = 99/200.
When x = 1, we have
1 / (2) (3) = 1/2 - 1/3
When x = 2, we have
1 / (3) (4) = 1/3 - 1/4
The rest is similar.
The sum of all these goes like this.
1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + 1/5 - 1/6 + 1/6 - 1/7 + 1/7 - 1/8 + ... + ... + 1/197 - 1/198 + 1/198 - 1/19 9 + 1/199 - 1/200
and you will see that there are many pairs cancelling out (1/3 against 1/3, 1/4 against 1/4 and so on, up to 1/199 against 1/199).
Surviving are the 1/2 and the -1/200, so combining these two, we get sum = 1/2 - 1/200 = 99/200.
360 Tutoring Program