Problem 71 - Precalculus: Mathematical for Calculus

Question

The following is a well-known children's rhyme: As I was going to St. Ives I met a man with seven wives; Every wife had seven sacks; Every sack had seven cats; Every cat had seven kits; Kits, cats, sacks, and wives, How many were going to St. Ives? Assuming that the entire group is actually going to St. Ives, show that the answer to the question in the rhyme is a partial sum of a geometric sequence, and find the sum.

Step-by-Step Solution

Answer

The narrator is going to St. Ives, not the group encountered.

Step 1: Understand the Problem

The problem describes a scenario where a man, accompanied by others, is headed to St. Ives. We need to determine the total number of individuals (kits, cats, sacks, and wives) going to St. Ives.

Step 2: Express the Sequence

Each wife has 7 sacks, each sack has 7 cats, and each cat has 7 kits. This creates a series of quantities: 7 wives, 7 sacks per wife, 7 cats per sack, and 7 kits per cat. Express these as powers of 7.

Step 3: Calculate Each Quantity

- Wives = 7 - Sacks = 7 (sacks per wife) × 7 (wives) = 7^2 - Cats = 7 (cats per sack) × 7 (sacks) × 7 (wives) = 7^3 - Kits = 7 (kits per cat) × 7 (cats) × 7 (sacks) × 7 (wives) = 7^4

Step 4: Calculate the Partial Sum of the Geometric Sequence

The number of members accompanying the man can be represented by a geometric sequence: 7 + 7^2 + 7^3 + 7^4, where the ratio of increase is 7 and the initial term (a) is 7.

Step 5: Find the Sum of the Geometric Sequence

The formula for the sum of a geometric series is given by \[ S_n = a \frac{r^n - 1}{r - 1} \]where \( a = 7 \), \( r = 7 \), and \( n = 4 \).Thus, \[ S_4 = 7 \frac{7^4 - 1}{7 - 1} = 7 \frac{2401 - 1}{6} = 7 \times 400 = 2800 \]

Step 6: Identify Individuals Actually Going to St. Ives

According to the riddle, only the narrator is definitely headed to St. Ives, while it does not explicitly state that this entire group is also going there.

Key Concepts

Sequence

A sequence in mathematics is simply an ordered list of numbers. Each number in the sequence is referred to as a term. Sequences are often represented using a format like: a_1, a_2, a_3, ..., a_n, where each "a" with a subscript represents each term in the sequence. In essence, sequences help us identify patterns, predict subsequent numbers, and often provide a basis for various mathematical concepts.
A sequence can be finite or infinite depending on the question or context. The rhyme in this exercise involves a specific, finite sequence involving the counting of wives, sacks, cats, and kits. Recognizing these sequences helps set the stage for solving more complex problems.
Sequences can be arithmetic, where each term increases by a constant difference, or geometric, where each term is multiplied by a constant factor. The rhyme in question involves a geometric sequence, which leads us to our next topic.

Geometric Sequence

When dealing with sequences, a geometric sequence stands out due to its unique pattern. In a geometric sequence, each term is found by multiplying the previous term by a constant value known as the "common ratio."
For example, in our problem, each group of quantities (wives, sacks, cats, and kits) multiplies by 7, the common ratio. Starting from the 7 wives, we proceed to 7^2 sacks, 7^3 cats, and 7^4 kits. The power of multiplication here becomes apparent as the numbers quickly grow larger.
Understanding geometric sequences is crucial because they form the basis for calculating sums, recognizing where each term comes from, and knowing the progression of values. This sequence's functionality leads directly into summing its terms, or finding the partial sum.

Partial Sum

The concept of a partial sum is vital when working with sequences, particularly when the sequence is geometric. A partial sum is the result of adding the first several terms of a sequence, rather than the entire sequence, which might be endless.
In the context of the rhyme, we calculate a partial sum of the geometric sequence to find the total companions of the man: the sum of 7, 7^2, 7^3, and 7^4. To achieve this, we use the geometric series sum formula: \[ S_n = a \frac{r^n - 1}{r - 1} \]where "a" is the first term (here, 7), "r" is the common ratio (also 7), and "n" is the number of terms (4 in our example).
Plugging the values into the formula gives us the partial sum: \[ S_4 = 7 \frac{7^4 - 1}{6} = 2800 \].
The exercise cleverly wraps up by challenging common assumptions directly posed by the rhyme. Although this calculated group wasn't confirmed to be going to St. Ives according to the riddle's traditional interpretation, the mathematical exercise is fruitful in demonstrating the application of partial sums in a geometric sequence.