Question

Angelina wins money in a competition. She is given the choice as to how she is paid.

Choice 1: Get $1 the 1st day, 2$ the 2nd day, $4 the 3rd day, 8$ the 4th day, and so on
Choice 2: Get 1,000,000 today.
a) With which method of payment will angelina get more money?
c) After how many days will the money Angelina gets from Choice 1 be approximately $1, 000, 000?

Answer 1

• Choice 1
• 20 days

Explanation:

You are being asked to compare the value of a growing infinite geometric series to a fixed constant. Such a series will always eventually have a sum that exceeds any given fixed constant.

__

a)

Angelina will get more money from the Choice 1 method of payment. The sequence of payments is a (growing) geometric sequence, so the payments and their sum will eventually exceed the alternative.

__

c)

For a first term of 1 and a common ratio of 2, the sum of n terms of the geometric series is given by ...

Sn = a1×(r^n -1)/(r -1) . . . . . . . . . . series with first term a1, common ratio r

We want to find n such that ...

Sn ≥ 1,000,000

1 × (2^n -1)/(2 -1) ≥ 1,000,000

2^n ≥ 1,000,001 . . . . add 1

n ≥ log(1,000,001)/log(2) . . . . . take the base-2 logarithm

n ≥ 19.93

The total Angelina receives from Choice 1 will exceed $1,000,000 after 20 days.