Question

MODELING WITH MATHEMATICS A marching band is arranged in rows. The fi rst row has three band members, and each row after the fi rst has two more band members than the row before it. (See Example 6.) a. Write a rule for the number of band members in the nth row. b. How many band members are in a formation with seven rows?

Short answer

a) The rule for the number of band members in the nth row is 1 + 2n. b) The total number of band members in a formation with seven rows is 56.

Step-by-step solution

Find the rule for the nth term

The number of band members increases by 2 with each row. This indicates arithmetic progression and according to the formula for arithmetic progression the nth term can be calculated as: a + (n-1) * d where 'a' is the first term, 'n' is the nth term we want to find and 'd' is the common difference. Here, we have a = 3 (number of members in first row), d = 2 (difference between the number of members in each row). Therefore, the nth term or the number of members in the nth row = 3 + (n-1) * 2.

Calculate the total number of members in seven rows

Here, we need to find the total sum of the members in seven rows. Sum of the terms of an arithmetic progression can be found using the formula: S_n = n/2 * (a + l) where 'n' is the number of terms (here it is number of rows), 'a' is the first term, and 'l' is the last term. The last term can be calculated from the formula derived in step 1 by substituting n = 7. Hence, the sum or total band members is found by substituting a = 3, l = 3 + (7-1) * 2, n = 7 in the sum formula.

Simplify and Calculate

We proceed to simplify and calculate the number of band members in the nth row and in seven rows according to the formulas in step 1 and 2. For the nth row: it simplifies to 1 + 2n. For seven rows: simplifying gives S_7 = 7/2 * (3 + 13), which then computes to 56.

Key concepts

Mathematical Modeling

Mathematical modeling is a fundamental process in mathematics and science that helps us create equations or formulas based on real-world scenarios.
It involves understanding a problem, identifying the variables, and then crafting mathematical expressions to describe relationships between these variables.
In our band example, mathematical modeling helps us determine how many members are in each row based on the given pattern. - We identify that the number of band members grows by two in each successive row, starting from the initial number of three in the first row. - By recognizing this pattern, we can create a rule or formula to model the situation mathematically. This rule takes the form of an arithmetic sequence formula, helping us predict the arrangement of the band members as the rows increase without having to physically count each one.
Mathematical modeling thus provides a structured way to generalize and solve various types of problems efficiently.

Sequences and Series

Sequences and series are two essential concepts in mathematics often used to represent lists of numbers following a specific pattern. A sequence is simply an ordered list of numbers, while a series is the sum of a sequence.
In the context of an arithmetic sequence, each term in the sequence is derived by adding a constant called the "common difference" to the previous term. - The sequence here is the number of band members in each row, which follows the pattern of increasing by two members per subsequent row. - The first term, or the first row, has three members. Then, each row adds two more, creating a sequence of numbers (3, 5, 7, 9, etc.). When you find the sum of the terms from a sequence, you are working with a series.
In our exercise, by summing all the band members from each row, we compute the total members needed for the full formation. By understanding sequences and series, we can easily deal with problems involving regular patterns, such as payment plans, scheduling, and more.

Arithmetic Sequence Formula

The arithmetic sequence is one of the most common types of sequences you'll encounter in mathematics.
Its defining feature is the constant addition of a fixed number, known as the common difference, to derive subsequent terms from the initial term.
The formula for finding any term in an arithmetic sequence is given by:\[ a_n = a + (n-1) \cdot d \]Where:- \(a\) is the first term of the sequence.- \(n\) is the term number you are trying to find.- \(d\) is the common difference between consecutive terms.In our band problem, the first term \(a\) is 3 (number of members in the first row), and the common difference \(d\) is 2 (the increase in the number of members per row).
This formula allows us to quickly determine the number of band members in any row without listing each one.Beyond singular term calculation, we can also use it to find the total sum of terms (or members, in this case) up to any given row with a series formula:\[ S_n = \frac{n}{2} \cdot (a + l) \]Where:- \(n\) is the number of terms.- \(l\) is the last term calculated using the arithmetic sequence formula.
These formulas serve as powerful tools in making predictions and confirming results in problems involving regular, predictable increases or decreases.