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# N represents the sum of reciprocals of consecutive integers from 301

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Joined: 07 Mar 2019
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N represents the sum of reciprocals of consecutive integers from 301  [#permalink]

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27 Nov 2019, 23:29
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Difficulty:

35% (medium)

Question Stats:

71% (01:28) correct 29% (02:23) wrong based on 14 sessions

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N represents the sum of reciprocals of consecutive integers from 301 to 400, inclusive. Choose the correct option?

a. 1/4 <N< 1/3

b. 1/6 <N< 1/4

c. 1/8 <N< 1/6

d. 1/10 <N< 1/8

e. 1/12 <N< 1/10

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Re: N represents the sum of reciprocals of consecutive integers from 301  [#permalink]

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27 Nov 2019, 23:36
lnm87 wrote:
N represents the sum of reciprocals of consecutive integers from 301 to 400, inclusive. Choose the correct option?

a. 1/4 <N< 1/3

b. 1/6 <N< 1/4

c. 1/8 <N< 1/6

d. 1/10 <N< 1/8

e. 1/12 <N< 1/10

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Re: N represents the sum of reciprocals of consecutive integers from 301  [#permalink]

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28 Nov 2019, 00:00
2
lnm87 wrote:
N represents the sum of reciprocals of consecutive integers from 301 to 400, inclusive. Choose the correct option?

a. 1/4 <N< 1/3

b. 1/6 <N< 1/4

c. 1/8 <N< 1/6

d. 1/10 <N< 1/8

e. 1/12 <N< 1/10

Method 1:

Given, $$N = \frac{1}{301} + \frac{1}{302} + \frac{1}{303} + . . . . . . . . . . . . + \frac{1}{400}$$

We Know that, $$\frac{1}{301} < \frac{1}{300}$$ and $$\frac{1}{302} < \frac{1}{300}$$ so on ...
--> $$N$$ is definitely less than $$\frac{1}{300} + \frac{1}{300} + \frac{1}{300} . . . . . . . . . . . . . . + \frac{1}{300}$$ (100 times)
--> $$N < 100*\frac{1}{300}$$
--> $$N < \frac{1}{3}$$

We Know that, $$\frac{1}{301} > \frac{1}{400}$$ and $$\frac{1}{302} > \frac{1}{400}$$ so on ...
--> $$N$$ is definitely greater than $$\frac{1}{400} + \frac{1}{400} + \frac{1}{400} . . . . . . . . . . . . . . + \frac{1}{400}$$ (100 times)
--> $$N > 100*\frac{1}{400}$$
--> $$N > \frac{1}{4}$$

--> $$\frac{1}{4}$$ < N < $$\frac{1}{3}$$

Method 2:

Formula: Sum of 'n' terms of Harmonic Progression is approximately = (number of terms)*(middle term) = $$100*\frac{1}{350} = \frac{1}{3.5}$$, which lies between $$\frac{1}{4}$$ & $$\frac{1}{3}$$

IMO Option A
Re: N represents the sum of reciprocals of consecutive integers from 301   [#permalink] 28 Nov 2019, 00:00
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