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Shortcut or Formula for sum of consecutive integers needed

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by Jeevanantham » Tue Jun 19, 2012 10:08 am
M is the sum of the reciprocals of the consecutive integers from 201 to 300, inclusive. Which of the following is true?

1. 1/3 < M < 1/2
2. 1/5 < M < 1/3
3. 1/7 < M < 1/5
4. 1/9 < M < 1/7
5. 1/12 < M < 1/9

Please explain this, also please tell the shortcut to find the sum of the consecutive integers.
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by Anurag@Gurome » Tue Jun 19, 2012 10:15 am
Jeevanantham wrote:M is the sum of the reciprocals of the consecutive integers from 201 to 300, inclusive. Which of the following is true?
M = (1/201 + 1/202 + 1/203 + ... + 1/299 + 1/300)

Note that there are 100 terms in the sum.
Also note that each term in the sum is less than 1/200 and more than 1/300 (except 1/300 which is equal to 1/300).

So, (1/300 + 1/300 + ... 100 times) < M < (1/200 + 1/200 + ... 100 times)
--> 100/300 < M < 100/200
--> 1/3 < M < 1/2

The correct answer is A.
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