Question

1) [10pts) Assume that the population of the world in 2010 was 6.9 billion and is growing at the rate of 1.1% a year. Find an explicit formula for the population of the world n years after 2010. Also, what will be the population of the world in 2030? Show all your work.

Answer 1

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Given: do = 6.9 billion r= 1.1% = 0.011 (a) Let an represents the population n years after 2010. Each year the population grows by 1.1%. Thus the population is the population of the previous year increased by 1.1% of the population of the previous year. an = an-1 + 1.1%. an-1 = an-1 + 0.011. an-1 = 1.011. an-1

(b) Given: an = 1.011.07-1 ao = 6.9 billion We successively apply the recurrence relation: an = 1.011an-1 = 1.011an-1 = 1.011(1.011a"-2) = 1.011’an-2 = 1.011²(1.011a-3) = 1.011an-3 = 1.011(1.0110"-2) = 1.011*an-4 = 1.011"anen = 1.011"ao = 6.9. 1.011" (c) Use the result of part (b) and evaluate it at n = 20 (since 2030 is 20 years after 2010): 8.6 020 = 6.9. 1.01120 Thus the population is 8.6 billion.

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