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?

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

Thus the common ratio is constant

This forms a geometric sequence

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

Where,

r is the common ratio,

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

So on for 15 days

Thus the sum is:

Thus total amount of this request is $ 327.67

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.