Respuesta :
The total amount of this request is $ 327.67
Solution:
Given that,
A friend asks you to borrow $.01 the first day, $.02 the second day, $.04 the third day, $.08 the fourth day, and so on for 15 days
Therefore, a sequence is formed as:
0.01, 0.02, 0.04, 0.08 , ....
Let us find the common ratio between terms
[tex]r = \frac{0.02}{0.01} = 2\\\\r = \frac{0.04}{0.02} = 2\\\\r = \frac{0.08}{0.04} = 2[/tex]
Thus the common ratio is constant
This forms a geometric sequence
The formula to find the first n terms of geometric sequence is:
[tex]S_n = \frac{a_1(1-r^n)}{1-r}[/tex]
Where,
r is the common ratio, [tex]r\neq 1[/tex]
[tex]S_n = sum\\\\a_1 = first\ term\\\\n = number\ of\ terms[/tex]
Here in 0.01, 0.02, 0.04, 0.08 , ....
So on for 15 days
[tex]a_1 = 0.01\\\\r = 2\\\\n = 15[/tex]
Thus the sum is:
[tex]S_{15} = \frac{0.01(1-2^{15})}{1-2}\\\\S_{15} = \frac{0.01(1-32768)}{-1}\\\\S_{15} = 0.01 \times 32767\\\\S_{15} = 327.67[/tex]
Thus total amount of this request is $ 327.67
