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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If S = 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2 + 1/8^2 + 1/9

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Math Expert V
Joined: 02 Sep 2009
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If S = 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2 + 1/8^2 + 1/9  [#permalink]

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Question Stats: 79% (01:12) correct 21% (01:26) wrong based on 1066 sessions

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If $$S = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \frac{1}{7^2} + \frac{1}{8^2} + \frac{1}{9^2} + \frac{1}{10^2}$$, which of the following is true?

A. S > 3

B. S = 3

C. 2 < S < 3

D. S = 2

E. S < 2

NEW question from GMAT® Official Guide 2019

(PS06243)

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GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4228
Re: If S = 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2 + 1/8^2 + 1/9  [#permalink]

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Top Contributor
6
Bunuel wrote:
If $$S = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \frac{1}{7^2} + \frac{1}{8^2} + \frac{1}{9^2} + \frac{1}{10^2}$$, which of the following is true?

A. S > 3

B. S = 3

C. 2 < S < 3

D. S = 2

E. S < 2

$$S = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \frac{1}{7^2} + \frac{1}{8^2} + \frac{1}{9^2} + \frac{1}{10^2}$$ = 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + . . . .

Now, let's convert a few of the fractions to decimal approximations...
S = 1 + 0.25 + 0.11 + 0.06 + 0.04 + . . .

S = 1.42 + 0.04 + . . .

IMPORTANT: Notice that each fraction is less than the fraction before it
So, all of the 5 decimals after 0.04 will be less than 0.04
So, if we replace all of those 5 decimals with 0.04, our new sum will be greater than the original sum

So: S < 1.42 + 0.04 + 0.04 + 0.04 + 0.04 + 0.04 + 0.04

Simplify: S < 1.42 + 0.24
Simplify: S < 1.66

Cheers,
Brent
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##### General Discussion
Manager  D
Joined: 17 May 2015
Posts: 241
If S = 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2 + 1/8^2 + 1/9  [#permalink]

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4
Bunuel wrote:
If $$S = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \frac{1}{7^2} + \frac{1}{8^2} + \frac{1}{9^2} + \frac{1}{10^2}$$, which of the following is true?

A. S > 3

B. S = 3

C. 2 < S < 3

D. S = 2

E. S < 2

NEW question from GMAT® Official Guide 2019

(PS06243)

Hi,

For simplicity, let's break the given series in three parts as follows:

$$S_{1} = 1$$,

$$S_{2} = \frac{1}{2^2} + \frac{1}{3^2}$$, and

$$S_{3} = \frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \frac{1}{7^2} + \frac{1}{8^2} + \frac{1}{9^2} + \frac{1}{10^2}$$

In $$S_{2}$$ larger term is $$\frac{1}{2^2}$$, and there are two terms. Hence,

$$S_{2} = \frac{1}{2^2} + \frac{1}{3^2} < 2*\frac{1}{4} = \frac{1}{2}$$ --- (1)

Similarly, in series $$S_{3}$$ the largest term is $$\frac{1}{4^2} = \frac{1}{16}$$ and there are total 7 terms. Hence,

$$S_{3} = \frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \frac{1}{7^2} + \frac{1}{8^2} + \frac{1}{9^2} + \frac{1}{10^2} < 7* \frac{1}{16} < 8*\frac{1}{16} = \frac{1}{2}$$ --- (2)

$$S = S_{1} + S_{2} + S_{3} < 1 + \frac{1}{2} + \frac{1}{2} = 2$$

Hence $$S < 2$$. Answer (E).

Thanks.
Marshall & McDonough Moderator D
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Re: If S = 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2 + 1/8^2 + 1/9  [#permalink]

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1
Since the terms after 1/9 add on to the hundreths place and since its easy to calculate upto 1/25, let's maximize all the terms after 1/25 by assuming the remaining terms as 1/25.

S = 1 + 0.25 + 0.11 + 0.0625 + 0.04*6
S = 1.4225 + 0.24 = 1.6625 < 2

Intern  B
Joined: 14 Feb 2018
Posts: 11
Location: Russian Federation
Re: If S = 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2 + 1/8^2 + 1/9  [#permalink]

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Can smbd post explanation from OG?
Director  D
Status: Learning stage
Joined: 01 Oct 2017
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WE: Supply Chain Management (Energy and Utilities)
If S = 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2 + 1/8^2 + 1/9  [#permalink]

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[quote="Bunuel"]If $$S = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \frac{1}{7^2} + \frac{1}{8^2} + \frac{1}{9^2} + \frac{1}{10^2}$$, which of the following is true?

A. S > 3

B. S = 3

C. 2 < S < 3

D. S = 2

E. S < 2

We have the terms after $$\frac{1}{5^2}$$ are close to zero. Hence the sum of the terms henceforth $$\frac{1}{5^2}$$ can be approximated to zero.
Hence $$S = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \frac{1}{7^2} + \frac{1}{8^2} + \frac{1}{9^2} + \frac{1}{10^2}$$ reduces to
$$S=1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2}$$
Or, $$S=\frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25}$$=1+0.25+0.11+0.0625+0.04=1.4625<2
So S<2.

Ans (E)
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PKN

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Intern  B
Joined: 09 May 2018
Posts: 8
Re: If S = 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2 + 1/8^2 + 1/9  [#permalink]

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I tried to solve it like below:
S= 1+ 1/2^2 + 1/3^2 + 1/4^2 + ...
S= 1 + {(1/6^2) * 9} [considering 1/6^2 as the the middle term in the remaining 9 terms. And multiplying it by number of terms. To get an approx value]
S= 1 + 1/4
S= 1.25
Hence (E)

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GMAT 1: 800 Q51 V51 If S = 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2 + 1/8^2 + 1/9  [#permalink]

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2
3
$$S = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \frac{1}{7^2} + \frac{1}{8^2} + \frac{1}{9^2} + \frac{1}{10^2}$$

GMAT quant questions tend to be designed to be answered efficiently via the use of hacks. So, let's start hacking.

Start off with the obvious: $$1 + \frac{1}{2^2} = 1 \frac{1}{4}$$

It becomes apparent that to get to 2 we need 3/4 more, and those other fractions look pretty small. They probably won't get us to 2.

If we can prove that 1 1/4 + (the sum of the rest of those fractions) < 2, then the correct answer will be S < 2 and we'll be done.

The next fraction is $$\frac{1}{3^2}$$.

That is effectively $$\frac{1}{9}$$.

We have 1 1/4 + 1/9.

Without adding, we know that 1 1/4 + 1/9 < 1 1/2.

The next seven are $$\frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \frac{1}{7^2} + \frac{1}{8^2} + \frac{1}{9^2} + \frac{1}{10^2}$$.

Since 8 x 1/16 = 1/2, and the first fraction of this set of seven fractions is 1/16 and all the rest are smaller, this set has to add up to less than 1/2.

So, S = 1 1/4 + 1/9 + (a number < 1/2).

S < 2.

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Intern  B
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Re: If S = 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2 + 1/8^2 + 1/9  [#permalink]

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1
NO Need of huge calculations done above.

Simple, the entire expression can be generalized as below:

S = 1+ 9*(1/n^2)
= n^2+9/n^2
so finally S = n^2+9/n^2

now put n=2 so S= 4 equation equals to 1.56 , put n=9 so 1.11

So, merely in all S< 2 so the option E is valid
Intern  B
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Re: If S = 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2 + 1/8^2 + 1/9  [#permalink]

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Easy way to solve - multiply both sides by 100 and then figure out rough estimation of S. Mine was ~1.6
Intern  Joined: 10 Aug 2019
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If S = 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2 + 1/8^2 + 1/9  [#permalink]

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Since 1/2^2=1/(2*2)<1/(1*2)=1-1/2
1/3^2<1/(2*3)=1/2-1/3

so S<1+(1-1/2)+(1/2-1/3)+...+(1/9-1/10)=2-1/10<2 If S = 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2 + 1/8^2 + 1/9   [#permalink] 18 Aug 2019, 20:28
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# If S = 1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2 + 1/8^2 + 1/9  