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Question 33 : An infinite geometric progression a_{1}, a_{2}, a_{3},... has the property that a_{n} = 3(a_{n+1} + a_{n+2} +....) for every n โฅ 1. If the sum a_{1} + a_{2} + a_{3} +...... = 32, then a_{5} is

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

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The sum up to infinity, a_{1} + a_{2} + a_{3} +...... = 32.

It can also be written as \\frac{๐}{1 โ ๐}) = 32.

We have been given that, a_{n} = 3(a_{n+1} + a_{n+2} +....) if n โฅ 1.

When n = 1, we get,

a = 3(a_{2} + a_{3} + .....)

โน a = \\frac{3๐๐}{1 โ ๐}) [Since, a_{2} + a_{3} +...... = \\frac{๐r}{1 โ ๐})]

โน 1 โ r = 3r

โน 1 = 4r

โน r = \\frac{1}{4})

Now finding the value of a is easy.

a = 32 ร (1 - \\frac{1}{4}))

โน a = 32 ร \\frac{3}{4})

โน a = 24

The value of a_{5} = ar^{4} , where a = 24 and r^{4} = (\\frac{1}{4}))^{4}

a_{5} = 24 ร \\frac{1}{4}) ร \\frac{1}{4}) ร \\frac{1}{4}) ร \\frac{1}{4})

Hence a_{5} = \\frac{3}{32})

The question is **"An infinite geometric progression a _{1}, a_{2}, a_{3},... has the property that a_{n} = 3(a_{n+1} + a_{n+2} +....) for every n โฅ 1. If the sum a_{1} + a_{2} + a_{3} +...... = 32, then a_{5} is" **

Choice C is the correct answer

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