Question

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

Short answer

Option A results in more money: 500500 compared to 524287 for option B.

Step-by-step solution

Calculate the total money for option A

To find the total money for option A, add the amounts received each day from day 1 to day 1000. The sequence is a decreasing arithmetic sequence starting at 1000 and decreasing by 1 each day. The sum of an arithmetic sequence can be found using the formula: \[ S = \frac{n}{2} (a + l) \] where \( n \) is the number of terms (1000), \( a \) is the first term (1000), and \( l \) is the last term (1). Thus \[ S_A = \frac{1000}{2} (1000 + 1) = 500 * 1001 = 500500 \].

Calculate the total money for option B

To find the total money for option B, add the amounts received each day from day 1 to day 19. This sequence is a geometric sequence starting at 1 and doubling each day. The sum of a geometric sequence can be found using the formula: \[ S = a \frac{r^n - 1}{r - 1} \] where \( a \) is the first term (1), \( r \) is the common ratio (2), and \( n \) is the number of terms (19). Thus \[ S_B = 1 \frac{2^{19} - 1}{2 - 1} = 2^{19} - 1 = 524288 - 1 = 524287 \].

Compare the total amounts

Compare the total amounts calculated for both options A and B. Option A: 500500Option B: 524287Since 524287 (B) is less than 500500 (A), option A results in more money.

Key concepts

arithmetic sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference, denoted as \( d \). The general form of an arithmetic sequence can be written as:

• \( a, a + d, a + 2d, a + 3d, \ldots \)

Here's an example with common difference \( d = 1 \): 1, 2, 3, 4, 5. Each term increases by 1.
To find the \( n^{th} \) term of an arithmetic sequence, you can use the formula: \( a_n = a + (n-1)d \). Here, \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference.
In the given exercise, option A is an arithmetic sequence starting at \( 1000 \) and ending at \( 1 \), with a common difference of \( -1 \). The sum of this series is found using the arithmetic sum formula: \[ S = \frac{n}{2} (a + l) \], where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term.

geometric sequence

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This sequence can be written as:

• \( a, ar, ar^2, ar^3,... \)

For example, if \( a = 2 \) and the common ratio \( r = 3 \), then the sequence is 2, 6, 18, 54, \ldots.

The \( n^{th} \) term of a geometric sequence can be calculated using \( a_n = ar^{n-1} \).
In the given exercise, option B is a geometric sequence starting at 1 and doubling each day for 19 days. The terms are: \( 1, 2, 4, 8, \ldots, 2^{18} \).
The sum of the first \( n \) terms of a geometric sequence is calculated using the formula: \[ S = a \frac{r^n - 1}{r - 1} \]. Here, \( a \) is 1, \( r \) is 2, and \( n \) is 19.

sum of series

The sum of a series refers to the sum of all terms up to a certain point. For arithmetic series, the formula to calculate the sum of the first \( n \) terms is:
\[ S = \frac{n}{2} (a + l) \]
For geometric series, the formula is:
\[ S = a \frac{r^n - 1}{r - 1} \].

Using these formulas, you can calculate the total amount given in a sequence. For instance, in the exercise, option A uses the arithmetic sum formula to find the total of \( 1000 + 999 + 998 + \ldots + 1 \), resulting in \( 500500 \). Option B uses the geometric sum formula for \( 1 + 2 + 4 + 8 \ldots + 2^{18} \), resulting in \( 524287 \).

comparison of sums

When comparing sums of different sequences, it's crucial to use the correct sum formulas and number of terms. In the exercise, we have:

• Option A: An arithmetic series of 1000 terms
• Option B: A geometric series of 19 terms

After calculating, we found:

• Sum of Option A: \( 500500 \)
• Sum of Option B: \( 524287 \)

Although the highest term in the arithmetic series seems significant, the geometric series grows exponentially, accumulating a large sum fast. Hence, comparisons must take growth rates and term counts into account to understand which is more beneficial.