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Question 33 : An infinite geometric progression a₁, a₂, a₃,... has the property that a_n = 3(a_n+1 + a_n+2 +....) for every n ≥ 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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Video Explanation

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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{𝑎}{1 − 𝑟}) = 32.
We have been given that, a_n = 3(a_n+1 + a_n+2 +....) if n ≥ 1.
When n = 1, we get,
a = 3(a₂ + a₃ + .....)
⟹ a = \\frac{3𝑎𝑟}{1 − 𝑟}) [Since, a₂ + a₃ +...... = \\frac{𝑎r}{1 − 𝑟})]
⟹ 1 – r = 3r
⟹ 1 = 4r
⟹ r = \\frac{1}{4})
Now finding the value of a is easy.
a = 32 × (1 - \\frac{1}{4}))
⟹ a = 32 × \\frac{3}{4})
⟹ a = 24
The value of a₅ = ar⁴ , where a = 24 and r⁴ = (\\frac{1}{4}))⁴
a₅ = 24 × \\frac{1}{4}) × \\frac{1}{4}) × \\frac{1}{4}) × \\frac{1}{4})
Hence a₅ = \\frac{3}{32})

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

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

Choice C is the correct answer