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Sequence S is the sequence of numbers a1, a2, a3, ... , an.

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Sequence S is the sequence of numbers a1, a2, a3, ... , an. [#permalink] New post 31 Jul 2020, 04:49
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Sequence S is the sequence of numbers \(a_1, a_2, a_3,\) ... , \(a_n\). For each positive integer n, the \(n\)th number \(a_n\) is defined by \(a_n=\frac{n+1}{3n}\) . What is the product of the first 53 numbers in sequence S?

A. \(\frac{2}{3^{53}}\)

B. \(\frac{2}{3^{50}}\)

C. \(\frac{2}{3^{49}}\)

D. \(\frac{3}{2^{50}}\)

E. \(\frac{2}{3^{25}}\)
[Reveal] Spoiler: OA

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Re: Sequence S is the sequence of numbers a1, a2, a3, ... , an. [#permalink] New post 31 Jul 2020, 04:54
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Re: Sequence S is the sequence of numbers a1, a2, a3, ... , an. [#permalink] New post 03 Aug 2020, 04:33
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an=(n+1)/3n

a1=2/3
a2=3/6
a3=4/9
a4=5/12
and so on

the product of all 53 terms will look like

=(2/3)*(3*6)*(4/9)*(5/12)*....
=(2/(3*1))*(3/(3*2))*(4/(3*3))*(5/(3*4))...
Notice that each of the 53 terms in the denominator we have a 3 as a factor. So when all these 3's get multiplied together we get

(2*3*4*5*...53*54)/((3^53)*1*2*3*4...*53)
The terms 2 to 53 get cancelled out with ones in the numerator leaving
=54/(3^53)
=(27*2)/(3^53)
=((3^3)*2)/(3^53)
=2/(3^50)

Final Answer: B
Re: Sequence S is the sequence of numbers a1, a2, a3, ... , an.   [#permalink] 03 Aug 2020, 04:33
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Sequence S is the sequence of numbers a1, a2, a3, ... , an.

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