You deposit $200 in a savings account.for each year thereafter , you plan to deposit 15% more than the previous year . About how much money will you have deposited in all after 20 years?

1 Answer
Jacob T.
Jun 17, 2018

$ color(white)(l) 20488.72$l20488.72

Explanation:

Amounts the person in question deposits each year

  • $ color(white)(l) 200$l200 in the first 1"st"1st year,
  • (1+15%) xx $ color(white)(l) 200(1+15%)×$l200 in the second 2"nd"2nd year,
  • (1+15%)^2 xx $ color(white)(l) 200(1+15%)2×$l200 in the third 3"rd"3rd year,
  • cdot cdot cdot⋅⋅⋅
  • (1+15%)^19 xx $ color(white)(l) 200(1+15%)19×$l200 in the twentieth 20"th"20th year,

form a geometric sequence.

A general formula gives the sum of the first n"th"nth terms of a geometric sequence of common ratio rr and first term a_1a1

sum_(i=1)^(n) r^(i-1) xx a_1 = a_1 xx (1-r^n)/(1-r)n∑i=1ri−1×a1=a1×1−rn1−r

The geometric sequence in this question has

r = 1+15% = 1.15r=1+15%=1.15

as its common ratio and

a_1=$ color(white)(l) 200a1=$l200

as the first term, which equals to the deposit in the very first year.

The question is asking for the sum of the first twentieth terms of this sequence, implying n=20n=20; substituting nn, rr, and a_1a1 with their respective values and evaluating the summation gives

sum_(i=1)^(20) 1.15^(i-1) xx $ color(white)(l) 200 = $ color(white)(l) 200 xx (1-1.15^20)/(1-1.15) = $ color(white)(l) 20488.7220∑i=11.15i−1×$l200=$l200×1−1.15201−1.15=$l20488.72
(rounded to the two decimal places)

Therefore the person would have deposited $ color(white)(l) 20488.72$l20488.72 in total in the twenty years.

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