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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

Explanation:

From the given data, populaiton in 

2019 : p

after 1 year i.e. in 2020 : 2p + 3

after 2 years i.e. in 2021 : 2(2p + 3) + 3 = 2²p + 2×3 + 3

after 3 years i.e. in 2022 : 2(2²p + 2×3 + 3) = 2³p + 2²×3 + 2×3 + 3

...

after n years i.e. : = 2ⁿp + 2^n-1×3 + 2^n-2×3 + ... + 3

Hence, population in 2034 i.e. after 15 years = 2¹⁵p + 2¹⁴×3 + 2¹³×3 + ... + 3

= 2¹⁵p + 3(2^n-1 + 2^n-2 + ... + 1)

= 2¹⁵p + 3(2¹⁵ - 1)/(2 - 1)

= 2¹⁵ × 1000 + 3(2¹⁵ - 1)

= 2¹⁵ × 1003 - 3

Hence, option (c).