Town Population Recurrence Relation

CAT 2019 Slot 1 · QA · Medium · Progressions

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If the population of a town is pp in the beginning of any year then it becomes 3+2p3+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

  1. A.

    (1003)15+6(1003)^{15} + 6

  2. B.

    (997)15−3(997)^{15} - 3

  3. C.

    (1003)215−3(1003)2^{15} - 3

  4. D.

    (997)214+3(997)2^{14} + 3

Answer

C

Explanation

Recurrence: pn+1=2pn+3p_{n+1} = 2p_n + 3. Add 33 to both sides: pn+1+3=2(pn+3)p_{n+1} + 3 = 2(p_n + 3). This forms a GP for (pn+3)(p_n + 3) with common ratio 22. Number of years from 2019 to 2034 is n=15n = 15. p2034+3=(p2019+3)⋅215=(1000+3)⋅215=1003⋅215p_{2034} + 3 = (p_{2019} + 3) \cdot 2^{15} = (1000 + 3) \cdot 2^{15} = 1003 \cdot 2^{15}. p2034=(1003)215−3p_{2034} = (1003)2^{15} - 3.

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