Problema Solution
How much money would you have all together after 20 days if you were given one dollar the first day, double that dollar the next day, then double the previous day's dollars each day?
Answer provided by our tutors
First day: d1 = 1
Second day: d2 = 2d1
Third day: d3 = 2*d2 = 2*2d1 = 2^2*d1
Fourth day: d4 = 2*d3 = 2^3*d1
......
Twentieth day: d20 = 2^19 d1
Altogether we need to find the sum of the first 20 members of a geometric sequence:
1 + 2d1 + 2^2d1 + .... + 2^19d1 = (1 + 2 + 2^2 + ... 2^19)d1 = (1 + 2 + 2^2 + ... 2^19)*1 = 1 + 2 + 2^2 + ... 2^19
For a geometric sequence, the sum of the first n terms with first term a1 = a and common ratio r, is given by the equation:
In our case n = 20, a = 1, r = 2 so we have
1 + 2 + 2^2 + ... 2^19 = 1(1 - 2^20)/(1 - 2) = 1,048,575
Altogether you would have $1,048,575.