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What is the value of 1/3C2 + 1/4C2 + 1/5C2 + … + 1/100C2?

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Math Revolution GMAT Instructor
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What is the value of 1/3C2 + 1/4C2 + 1/5C2 + … + 1/100C2?  [#permalink]

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New post 29 Mar 2019, 00:19
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A
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C
D
E

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  45% (medium)

Question Stats:

50% (01:26) correct 50% (01:23) wrong based on 10 sessions

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[GMAT math practice question]

What is the value of 1/3C2 + 1/4C2 + 1/5C2 + … + 1/100C2?

\(A. \frac{49}{50}\)
\(B. \frac{51}{50}\)
\(C. \frac{99}{100}\)
\(D. \frac{101}{100}\)
\(E. \frac{49}{100}\)

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What is the value of 1/3C2 + 1/4C2 + 1/5C2 + … + 1/100C2?  [#permalink]

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New post 29 Mar 2019, 06:22
1
MathRevolution wrote:
[GMAT math practice question]

What is the value of 1/3C2 + 1/4C2 + 1/5C2 + … + 1/100C2?

\(A. \frac{49}{50}\)
\(B. \frac{51}{50}\)
\(C. \frac{99}{100}\)
\(D. \frac{101}{100}\)
\(E. \frac{49}{100}\)


\(\frac{1}{3C2}+\frac{1}{4C2}+\frac{1}{5C2}+.......\frac{1}{100C2}\)..
\(\frac{2}{3*2}+\frac{2}{4*3}+\frac{2}{5*4}+.......\frac{2}{100*99}\)..
\(2(\frac{1}{3*2}+\frac{1}{4*3}+\frac{1}{5*4}+.......\frac{1}{100*99})\)..
\(2((\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+(\frac{1}{4}-\frac{1}{5})+.......(\frac{1}{98}-\frac{1}{99})+(\frac{1}{99}-\frac{1}{100}))\)..
\(2(\frac{1}{2}-\frac{1}{100})=2*\frac{49}{100}\)..
\(\frac{49}{50}\)..
A
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Re: What is the value of 1/3C2 + 1/4C2 + 1/5C2 + … + 1/100C2?  [#permalink]

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New post 31 Mar 2019, 18:33
=>

nCr = \(\frac{{ n(n-1)…(n-r+1) }}{{ 1*2*… *r }}\)
So, nC2 = \(\frac{n(n-1)}{2}\) and 1/nC2 = \(\frac{2}{{n(n-1)}}\).
Recall that \(\frac{1}{{k(k+1)}} = \frac{1}{k} – \frac{1}{(k+1)}.\)
So,
1/3C2 + 1/4C2 + 1/5C2 + … + 1/100C2
\(= \frac{2}{(2*3)} + \frac{2}{(3*4)} + \frac{2}{(4*5)} + … + \frac{2}{(99*100)}\)
\(= 2{ \frac{1}{(2*3)} + \frac{1}{(3*4)} + \frac{1}{(4*5)} + … + \frac{1}{(99*100)} }\)
\(= 2{ ( \frac{1}{2} – \frac{1}{3} ) + ( \frac{1}{3} – \frac{1}{4} ) + ( \frac{1}{4} – \frac{1}{5} ) + … + ( \frac{1}{99} – \frac{1}{100} ) }\)
\(= 2( \frac{1}{2} – \frac{1}{100} )\)
\(= 2 ( \frac{50}{100} – \frac{1}{100} )\)
\(= 2(\frac{49}{100})\)
\(= \frac{49}{50}\)

Therefore, the answer is A.
Answer: A
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Re: What is the value of 1/3C2 + 1/4C2 + 1/5C2 + … + 1/100C2?   [#permalink] 31 Mar 2019, 18:33
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