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CAT 2019 Slot 1 QA Question & Solution

AlgebraMedium

Question

If the population of a town is p in the beginning of any year then it becomes 3 + 2p in the beginning of the next year. If the population in the beginning of 2019 is 1000, then the population in the beginning of 2034 will be

Options

$(1003)^{15} + 6$
$(997)^{15} - 3$
$(997)2^{14} + 3$
$(1003)2^{15} - 3$

Solution

The population of town at the beginning of 1st year = p

The population of town at the beginning of 2nd year = 3+2p

The population of town at the beginning of 3rd year = 2(3+2p)+3 = 2*2p+2*3+3 =4p+3(1+2)

The population of town at the beginning of 4th year = 2(2*2p+2*3+3)+3 = 8p+3(1+2+4)

Similarly population at the beginning of the nth year = $2^{n-1}$p+3($2^{n-1}-1$) = $2^{n-1}\left(p+3\right)$-3 

The population in the beginning of 2019 is 1000, then the population in the beginning of 2034 will be $(2^{2034-2019})\left(1000+3\right)$-3 = $2^{15}\left(1003\right)$-3