Suppose you go to work for a company that pays one penny on the first day, 2 cents on the second day, 4 cents on the third day and so on.

Question

Suppose you go to work for a company that pays one penny on the first day, 2 cents on the second day, 4 cents on the third day and so on.

Hint: use an= a1 (r)^n-1 and Sn= a1 (1-r^n) / 1 - r

A. If the daily wage keeps doubling, what would your income be on day 31? Give your answer in dollars NOT pennies.

Income on day 31 = $ __________

B. If the daily wage keeps doubling, what will your total income be for working 31 days? Give your answer in dollars NOT pennies.

Total Income for working 31 days = $ _________

Answer 1

Hello, thank you for taking the time to post your question!

This question follows a geometric sequence, meaning that

an = a1(r)^(n-1) will be the underlying formula here

for the scenario in part (a) we’re using r = 2, n = 31, and a1 = 1 and then solving for a31

a31 = 1(2)^(31-1) = 1(2)^30 = 2^30 pennies

converting that over to dollars gives 2^30 / 100 = $10,737,418.24 for the income on Day 31, which is astoundingly large if you think about it

for part (b) then it’s after the sum of up to 31 days … so that would mean taking

Sn = a1(1 – r^n) / (1 – r)

Sn = (2^31 – 1)/100 = $21,474,836.47

I hope that helps get you moving in the right direction! Feel free to reach out if you have any additional questions beyond that :)

Answer 2

This is a typical Geometric Series because the pattern described (1, 2, 4, ...) is a series where we are multiplying by a rate to give us the following term. In this case, our rate is 2, since we are doubling our money each day (r=2). a₁ represents the first term in the series, in this case it is 1. a_n represents the n^th term in the series. Lastly, S_n represents the sum of the series. In this example, this represents the total money made at the company.

Part A. Find a_n using a_n=a₁(r)^n-1 where n=31, a₁=1, and r=2

Part B. Find S_n using S_n= a₁(1-rⁿ)/(1-r) where n=31, a₁=1, and r=2

Hope this helps!