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?

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.