Problem 59 - Algebra 2

Question

Evaluate each series to the given term. \(12.5+15+17.5+20+22.5+\ldots ; 7th\) term

Step-by-Step Solution

Answer

The 7th term of the series is 27.5

Step 1: Identify the first term, \(a_1\)

In this series, the first term, represented as \(a_1\), is 12.5.

Step 2: Identify the common difference, \(d\)

The common difference \(d\) in this arithmetic progression can be found by subtracting the second term from the first term. The calculation is \(15 - 12.5 = 2.5\). So, \(d = 2.5\)

Step 3: Use the formula to calculate \(a_n\)

Plug the values we found into the formula and calculate: \(a_n = a_1 + (n-1) * d \) becomes \(a_7 = 12.5 + (7-1) * 2.5 \). After calculation, the 7th term is 27.5

Key Concepts

Common Difference

In an arithmetic series, the common difference, denoted as **d**, is the amount by which consecutive terms increase or decrease. This value is constant throughout the progression. To find this common difference, subtract any term in the sequence from the subsequent term. For example, in the series given, the second term is 15 and the first term is 12.5. Subtracting these, you find the common difference:

• \( d = 15 - 12.5 = 2.5 \)

This tells us that each term in the sequence increases by 2.5 from the previous term. Knowing the common difference helps in predicting future terms and understanding the pattern of the sequence. Without it, the arithmetic nature of the sequence could not be maintained.

First Term

The first term, often denoted as **a₁**, is the starting point of an arithmetic sequence. It's essential because every other term in the sequence is derived from this initial value by adding the common difference repeatedly. In this example, the first term is 12.5.

• The series begins at 12.5

This initial term lays the foundation for calculating any other term in the series. Understanding this is critical because it sets the stage for the entire sequence. Without this value, you cannot accurately determine any terms that follow.

Nth Term

The nth term in an arithmetic sequence is a formula to find any term in the sequence without listing all the previous terms. The formula generally used is: \[ a_n = a_1 + (n-1) \times d \]
In our exercise, to find the 7th term (\( a_7 \)), you substitute the known values:

• \( a_1 = 12.5 \)
• \( d = 2.5 \)
• \( n = 7 \)

Plug these into the formula:

• \( a_7 = 12.5 + (7-1) \times 2.5 \)
• \( = 12.5 + 6 \times 2.5 \)
• \( = 12.5 + 15 \)
• \( = 27.5 \)

Thus, the 7th term is 27.5. This shows how powerful the formula is in quickly finding terms of the sequence.

Sequence Calculation

Sequence calculation in an arithmetic series builds on understanding the first term and common difference. To find successive terms, use these values in conjunction with the formula for the nth term. This process enables predicting any term in the series without directly calculating each term sequentially.
For instance, with a first term of 12.5 and a common difference of 2.5, the sequence continues as follows:

• 1st term: 12.5
• 2nd term: 15
• 3rd term: 17.5
• 4th term: 20
• 5th term: 22.5
• 6th term: 25
• 7th term: 27.5

Using the nth-term formula streamlines this calculation process and avoids unnecessary intermediate steps. It's a helpful strategy that adds efficiency and precision to solving sequences.