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CAT Previous Year Paper | CAT Progressions Questions | Question 33

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Question 33 : An infinite geometric progression a1, a2, a3,... has the property that an = 3(an+1 + an+2 +....) for every n โ‰ฅ 1. If the sum a1 + a2 + a3 +...... = 32, then a5 is

  1. \\frac{1}{32})
  2. \\frac{2}{32})
  3. \\frac{3}{32})
  4. \\frac{4}{32})


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Explanatory Answer

Method of solving this CAT Question from Progressions

The sum up to infinity, a1 + a2 + a3 +...... = 32.
It can also be written as \\frac{๐‘Ž}{1 โˆ’ ๐‘Ÿ}) = 32.
We have been given that, an = 3(an+1 + an+2 +....) if n โ‰ฅ 1.
When n = 1, we get,
a = 3(a2 + a3 + .....)
โŸน a = \\frac{3๐‘Ž๐‘Ÿ}{1 โˆ’ ๐‘Ÿ}) [Since, a2 + a3 +...... = \\frac{๐‘Žr}{1 โˆ’ ๐‘Ÿ})]
โŸน 1 โ€“ r = 3r
โŸน 1 = 4r
โŸน r = \\frac{1}{4})
Now finding the value of a is easy.
a = 32 ร— (1 - \\frac{1}{4}))
โŸน a = 32 ร— \\frac{3}{4})
โŸน a = 24
The value of a5 = ar4 , where a = 24 and r4 = (\\frac{1}{4}))4
a5 = 24 ร— \\frac{1}{4}) ร— \\frac{1}{4}) ร— \\frac{1}{4}) ร— \\frac{1}{4})
Hence a5 = \\frac{3}{32})

The question is "An infinite geometric progression a1, a2, a3,... has the property that an = 3(an+1 + an+2 +....) for every n โ‰ฅ 1. If the sum a1 + a2 + a3 +...... = 32, then a5 is"

Hence, the answer is \\frac{3}{32})

Choice C is the correct answer

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