Question

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?

Answer 1

`$ color(white)(l) 20488.72`

Explanation:

Amounts the person in question deposits each year

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

form a geometric sequence.

A general formula gives the sum of the first `n"th"` terms of a geometric sequence of common ratio `r` and first term `a_1`

`sum_(i=1)^(n) r^(i-1) xx a_1 = a_1 xx (1-r^n)/(1-r)`

The geometric sequence in this question has

`r = 1+15% = 1.15`

as its common ratio and

`a_1=$ color(white)(l) 200`

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=20`; substituting `n`, `r`, and `a_1` 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) 20488.72`
(rounded to the two decimal places)

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