# CAT 2017 Question Paper | Quants Slot 2

###### 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 ≥ 1. If the sum a1 + a2 + a3 +...... = 32, then a5 is

1. $$frac{1}{32}$ 2. $\frac{2}{32}$ 3. $\frac{3}{32}$ 4. $\frac{4}{32}$ ## 🎉 New Year offer! Fabulous 4,000 off on CAT Online Courses Valid until 15th April #### 2IIM : Best Online CAT Coaching. ### Video Explanation ## Best CAT Coaching in Chennai #### CAT Coaching in Chennai - CAT 2021Online Batches Available Now! ### Explanatory Answer ##### Method of solving this CAT Question from Progressions The sum up to infinity, a1 + a2 + a3 +...... = 32. It can also be written as $\frac{𝑎}{1 − 𝑟}$ = 32. We have been given that, an = 3$an+1 + an+2 +....) if n ≥ 1.
When n = 1, we get,
a = 3(a2 + a3 + .....)
⟹ a = $$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 an = 3$an+1 + an+2 +....) for every n ≥ 1. If the sum a1 + a2 + a3 +...... = 32, then a5 is"

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