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655-705 (Hard)|   Sequences|      
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Bunuel
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For the sequence a1, a2 … an, every term after the first can be found 08 Feb 2022, 09:45
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E
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55% (hard)

Question Stats:

71% (02:20) correct 29% (02:27) wrong based on 381 sessions

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For the sequence \(a_1\), \(a_2\), …, an, every term after the first can be found using the equation \(a_n=2(a_{n−1})\). If \(a_1=\frac{1}{2}\), then \(a_{49}\) is what percent less than \(a_{52}\)?

A. 12.5
B. 25
C. 50
D. 75
E. 87.5


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E
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rajatchopra1994
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For the sequence a1, a2 … an, every term after the first can be found 08 Feb 2022, 10:36
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Explanation:

a1 = 1/2
a2 = 1
a3 = 2
a4 = 4 = 2^2
a5 = 8 = 2^3
So, a49 = 2^47
a52 = 2^50

% Less = (2^50 - 2^47)×100/(2^50)
= 2^47×(2^3 - 1)×100 / 2^47×2^3
= 7×100/8
= 87.5

IMO-E

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For the sequence a1, a2 … an, every term after the first can be found 08 Feb 2022, 11:53
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Bunuel
For the sequence \(a_1\), \(a_2\), …, an, every term after the first can be found using the equation \(a_n=2(a_{n−1})\). If \(a_1=\frac{1}{2}\), then \(a_{49}\) is what percent less than \(a_{52}\)?

A. 12.5
B. 25
C. 50
D. 75
E. 87.5


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We know: \(a_n=2(a_{n−1})\)

Thus, each term is 2 times the preceding term
Thus, if the 49th term be p, we have:
50th term = 2p
51st term = 4p
52nd term = 8p

Thus, the 49th term is less than the 52nd term by: [(8p - p)/8p] x 100 = (7/8) x 100 = 87.5%

Note: The value of the 1st term is not necessary to answer the question

Answer E
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For the sequence a1, a2 … an, every term after the first can be found 17 Apr 2023, 14:29
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Bunuel
For the sequence \(a_1\), \(a_2\), …, an, every term after the first can be found using the equation \(a_n=2(a_{n−1})\). If \(a_1=\frac{1}{2}\), then \(a_{49}\) is what percent less than \(a_{52}\)?

A. 12.5
B. 25
C. 50
D. 75
E. 87.5


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We are told that each term in this sequence is double the previous one, therefore this is a geometric progression with a ratio r=2.
Using this fact; A52= A49x2x2x2= A49x8

We are asked A49 is what percent less than A52; which can be translated as: A49= (100%-x%) A52
A49/A52= 100%-x%
A49/A49x8= 1/8=100%-x%

We must memorize that 1/8= 12,5%
Hence x%=100%-12,5%= 87,5%

Correct answer is E
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For the sequence a1, a2 … an, every term after the first can be found 17 Apr 2023, 22:31
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Bunuel
For the sequence \(a_1\), \(a_2\), …, an, every term after the first can be found using the equation \(a_n=2(a_{n−1})\). If \(a_1=\frac{1}{2}\), then \(a_{49}\) is what percent less than \(a_{52}\)?

A. 12.5
B. 25
C. 50
D. 75
E. 87.5
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Solution:

  • The given sequence is a GP with first term \(a=\frac{1}{2}\) and common ratio \(r=2\)

  • We are asked the value of \(\frac{a_{52}-a_{49}}{a_{52}}\times 100\)
    \(=\frac{a\times r^{51}-a\times r^{48}}{a\times r^{51}}\times 100\) (General term of GP \(=ar^{n-1}\))

    \(=\frac{\frac{1}{2}\times 2^{51}-\frac{1}{2}\times 2^{48}}{\frac{1}{2}\times 2^{51}}\times 100\)

    \(=\frac{2^{50}-2^{47}}{2^{50}}\times 100\)

    \(=(1-\frac{1}{8})\times 100\)

    \(=\frac{7}{8}\times 100\)

    \(=87.5\%\)

Hence the right answer is Option E
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For the sequence a1, a2 … an, every term after the first can be found 08 Oct 2025, 00:17
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