The first 2 terms of a geometric progression are 3 and -2. Write down a simplified expression for the sum to n terms and evaluate the sum to infinity of this progression.
Find the least value of n for the sum to n terms to be within 2% of the sum to infinity.
In this case, I shall denote the sum to infinity as 'i' and the sum to n terms as 'S'
From my understanding, I'd interpret the last statement of the question as
| S | </= 0.02 i, since the question specifies that it be within 2%, meaning if the 'S' is 2% of i, it if still valid.
However, the solutions provided gave something like
| S - i | < 0.02 i
and they solved accordingly, getting n = 10 as the answer. If I used my own method I get n = 7. I wonder why they subtract 'i' from 'S'. And why is the difference LESS than 0.02i and not LESS THAN OR EQUAL TO 0.02i?
Thank you.
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Originally posted by Audi:The first 2 terms of a geometric progression are 3 and -2. Write down a simplified expression for the sum to n terms and evaluate the sum to infinity of this progression.
Find the least value of n for the sum to n terms to be within 2% of the sum to infinity.
In this case, I shall denote the sum to infinity as 'i' and the sum to n terms as 'S'
From my understanding, I'd interpret the last statement of the question as
| S | </= 0.02 i, since the question specifies that it be within 2%, meaning if the 'S' is 2% of i, it if still valid.
However, the solutions provided gave something like
| S - i | < 0.02 i
and they solved accordingly, getting n = 10 as the answer. If I used my own method I get n = 7. I wonder why they subtract 'i' from 'S'. And why is the difference LESS than 0.02i and not LESS THAN OR EQUAL TO 0.02i?
Thank you.
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http://www.thefreedictionary.com/within
"within" means inside.
I don't think any Maths explanation is needed here, so here goes:
Imagine a city with a fence around it .
< means inside the city
= means on the fence
> means outside the city
Within is inside, so it does not include being on the fence.
I hope this analogy helps.
Hi Audi,
This type of question is very common in which the topic of AP/GP is assessed along with the topic of inequalities.
In a GP, we observe that S_n (sum of n terms) is typically smaller than S (sum to infinity). The size of this difference involves the modulus and is |S - S_n|, since size is always a positive value. Therefore it makes logical sense for one to ask for the least value of n such that the size meets the condition of a certain tolerance, i.e., some percentage of S.
In short, we want to solve for n such that
size of difference (associated with the modulus) <= tolerance.
Thanks.
Cheers,
Wen Shih