The total amount of this request is $ 327.67
<em><u>Solution:</u></em>
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 , ....
<em><u>Let us find the common ratio between terms</u></em>
![r = \frac{0.02}{0.01} = 2\\\\r = \frac{0.04}{0.02} = 2\\\\r = \frac{0.08}{0.04} = 2](https://tex.z-dn.net/?f=r%20%3D%20%5Cfrac%7B0.02%7D%7B0.01%7D%20%3D%202%5C%5C%5C%5Cr%20%3D%20%5Cfrac%7B0.04%7D%7B0.02%7D%20%3D%202%5C%5C%5C%5Cr%20%3D%20%5Cfrac%7B0.08%7D%7B0.04%7D%20%3D%202)
Thus the common ratio is constant
This forms a geometric sequence
<em><u>The formula to find the first n terms of geometric sequence is:</u></em>
![S_n = \frac{a_1(1-r^n)}{1-r}](https://tex.z-dn.net/?f=S_n%20%3D%20%5Cfrac%7Ba_1%281-r%5En%29%7D%7B1-r%7D)
Where,
r is the common ratio, ![r\neq 1](https://tex.z-dn.net/?f=r%5Cneq%201)
![S_n = sum\\\\a_1 = first\ term\\\\n = number\ of\ terms](https://tex.z-dn.net/?f=S_n%20%3D%20sum%5C%5C%5C%5Ca_1%20%3D%20first%5C%20term%5C%5C%5C%5Cn%20%3D%20number%5C%20of%5C%20terms)
Here in 0.01, 0.02, 0.04, 0.08 , ....
So on for 15 days
![a_1 = 0.01\\\\r = 2\\\\n = 15](https://tex.z-dn.net/?f=a_1%20%3D%200.01%5C%5C%5C%5Cr%20%3D%202%5C%5C%5C%5Cn%20%3D%2015)
<em><u>Thus the sum is:</u></em>
![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](https://tex.z-dn.net/?f=S_%7B15%7D%20%3D%20%5Cfrac%7B0.01%281-2%5E%7B15%7D%29%7D%7B1-2%7D%5C%5C%5C%5CS_%7B15%7D%20%3D%20%5Cfrac%7B0.01%281-32768%29%7D%7B-1%7D%5C%5C%5C%5CS_%7B15%7D%20%3D%200.01%20%5Ctimes%2032767%5C%5C%5C%5CS_%7B15%7D%20%3D%20327.67)
Thus total amount of this request is $ 327.67