CAT 2017 Question Paper | Quants Slot 2

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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. $\frac{4}{32}$

Video Explanation

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

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