Critical Thinking Which of the following choices, A or B,...

Critical Thinking Which of the following choices, $A$ or $B$, results in more money? $A$ : To receive $$\$ 1000$$ on day $$1, \$ 999$$ on day $$2, \$ 998$$ on day 3 , with the process to end after 1000 days $B$ : To receive $$\$ 1$$ on day $$1, \$ 2$$ on day $$2, \$ 4$$ on day 3 , for 19 days

Critical Thinking Which of the following choices, $A$ or $B$, results in more money?
$A$ : To receive $$\$ 1000$$ on day $$1, \$ 999$$ on day $$2, \$ 998$$ on day 3 , with the process to end after 1000 days
$B$ : To receive $$\$ 1$$ on day $$1, \$ 2$$ on day $$2, \$ 4$$ on day 3 , for 19 days

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which each term is obtained by adding a constant value, known as the common difference, to the previous term. The total sum of the series can be computed using formulas that depend on the number of terms, the first term, and the last term or common difference. This concept is key when examining situations where increments or decrements occur uniformly over time.

Geometric Series

A geometric series is a sequence where each term is derived by multiplying the previous term by a fixed constant called the common ratio. The sum of a geometric series is calculated using a specific formula that takes into account the first term, the common ratio, and the total number of terms. This concept is essential in understanding phenomena that involve multiplicative growth or decay over time.

Exponential Growth

Exponential growth refers to an increase that occurs by a fixed multiplicative rate over equal intervals, often leading to rapid increases compared to linear (arithmetic) growth. This concept is crucial for understanding cases where values multiply rather than add, especially in scenarios involving processes like doubling.

Transcript

00:01 We have two choices which can get us a lot of money, which will get us more money.

00:04 Well, to determine this, let's model them both as sequences.

00:08 So first off, i'm going to be starting by modeling the top one.

00:11 This looks like an arithmetic sequence.

00:14 That is, when you go from 1 ,000 to 99, that's a difference of minus 1.

00:21 And 999 to 998 also has a difference of minus 1.

00:27 As it was minus one both times we have a common difference of negative one all right so this is going to be going for a thousand days so we can say that n equals 1 ,000 and we can say that our initial term a 1 is equal to 1 ,000 so a 1 ,000 that is our 1 ,000 day term will just be 1 ,000 minus 999 which is going to be be 1.

01:00 Because on our first day we had 1 ,000 minus 0, we could model this as 1 ,000 minus and minus 1, where n goes from 1 to 1 ,000.

01:15 Therefore, on our 1 ,000 day, we'll have 1 ,000 minus 1 ,000 minus 1, which will give us just $1 on the last day.

01:26 So with all of this in mind, the the summation of the first 50 terms of an arithmetic sequence, this is given in a formula in the book, is equal to n over 2, that is 1 ,000 over 2, times a1, 1 ,000 plus 1.

01:43 All right, and then if you multiply this out, it's not very difficult, you will get that this is equal to $500 ,000 ,500 at the end of this period.

01:53 So that's how much money we make for the first one.

01:55 Now how about the second one? well, this one is going to be a geometric sequence...

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