algebra word problem

Question

a city was incorporated in 2004 with a population of 35000. it is expected that the population will increase at a rate of 2% per year. the population n years after 2004 is given by the sequence Pn= 35000(1.02)^n

 

a. fin the first five terms of the sequence

b. find the population in 2014

Answer 1

a. 1.02^n is given by the binomial expansion of (1+x)^n where x=0.02: 1+nx+n(n-1)x^2/2+n(n-1)(n-2)x^3/6+n(n-1)(n-2)(n-3)x^4/24 to the first 5 terms. If we replace x by 0.02 we get: 1+0.02n+0.0002n(n-1)+0.000004n(n-1)(n-2)/3+0.00000002(n-1)(n-2)(n-3)/3.

b. 35000*1.02^(2014-2004)=35000*1.02^10=42665 approx. Using the binomial expansion:

35000(1+0.2+0.018+0.00096+0.0000336)=35000*1.2189936=42664.776 or 42665 approx.