Problem 82 - A Graphical Approach to Precalculus with Limits

Question

A stack of telephone poles has 30 in the bottom row, 29 in the next, and so on, with one pole in the top row. How many poles are in the stack?

Step-by-Step Solution

Answer

There are 465 poles in the stack.

Step 1: Identify the Sequence

The sequence starts with 30 poles in the bottom row and decreases by one pole each row up to the single pole at the top row. This is an arithmetic sequence where the first term is 30 and the last term is 1.

Step 2: Determine the Number of Terms

The number of terms in the sequence is the same as the number of rows, which is from 30 to 1, inclusive. Since the sequence decreases from 30 to 1, there are 30 terms.

Step 3: Use the Formula for the Sum of an Arithmetic Sequence

The formula for the sum of an arithmetic sequence is \( S = \frac{n}{2} (a + l) \), where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term. Here, \( n = 30 \), \( a = 30 \), and \( l = 1 \).

Step 4: Calculate the Total Number of Poles

Using the formula \( S = \frac{30}{2} (30 + 1) \), calculate the sum: \[ S = 15 \times 31 = 465 \].

Step 5: Confirm the Result

Double-check the calculations: \( 15 \times 31 \) equals 465, confirming the total number of poles.

Key Concepts

Sum of Arithmetic Sequence

The sum of an arithmetic sequence involves adding up all the terms within the sequence. To find the total sum, we use a specific formula designed for arithmetic sequences. This formula helps in calculating the sum efficiently without the need to manually add each term.

For an arithmetic sequence, the sum is calculated using:

• \( S = \frac{n}{2} (a + l) \)

Here, \( S \) represents the sum of the sequence, \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term.
This formula works by taking the average of the first and last term and then multiplying it by the number of terms. This average gives us the middle point of the sequence, and since the average stays constant, multiplying by the total number of terms gives the whole sum.

Understanding this formula is key to solving arithmetic sequence problems quickly and effectively.

Arithmetic Sequence Formula

The arithmetic sequence formula is crucial for identifying and working with sequences where each term increases or decreases by a constant amount. In this problem, the sequence of telephone poles forms an arithmetic sequence.

Each term in an arithmetic sequence is given by:

• \( a_n = a_1 + (n-1)\cdot d \)

Where:

• \( a_n \) is the \( n \)-th term
• \( a_1 \) is the first term
• \( n \) is the term number
• \( d \) is the common difference between the terms

In our exercise, the sequence starts at 30 and reduces by 1 each time, thus \( d = -1 \). Understanding how to manipulate and apply this formula lets you determine any specified term's value within the sequence.

This calculation is essential when solving or verifying the correctness of your sequence-related problems.

Number of Terms in a Sequence

Knowing the number of terms in a sequence is important because it directly impacts how we use the sum formula. Find the number of terms by evaluating the sequence's start and end points.

To find the number of terms, consider:

• The sequence's progression, which in this case was decrementing by 1 from 30 to 1
• Every distinct term from 30 down to 1 is a unique term in this sequence

Ultimately, there are exactly 30 terms.

Understanding how many terms exist allows us to apply the sum formula in solving for the total number of items (telephone poles, in this case) efficiently. Thus knowing and accurately counting the terms empowers us to solve many sequence problems.