Question

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. What is the total amount of this request?

Answer 1

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

$r = \frac{0.02}{0.01} = 2\\\\r = \frac{0.04}{0.02} = 2\\\\r = \frac{0.08}{0.04} = 2$

Thus the common ratio is constant

This forms a geometric sequence

The formula to find the first n terms of geometric sequence is:

$S_n = \frac{a_1(1-r^n)}{1-r}$

Where,

r is the common ratio, $r\neq 1$

$S_n = sum\\\\a_1 = first\ term\\\\n = number\ of\ terms$

Here in 0.01, 0.02, 0.04, 0.08 , ....

So on for 15 days

$a_1 = 0.01\\\\r = 2\\\\n = 15$

Thus the sum is:

$S_{15} = \frac{0.01(1-2^{15})}{1-2}\\\\S_{15} = \frac{0.01(1-32768)}{-1}\\\\S_{15} = 0.01 \times 32767\\\\S_{15} = 327.67$

Thus total amount of this request is $ 327.67