Problem 73 - Algebra for College Students

Question

Explain in words how to find the sum of the first \(n\) terms of an arithmetic sequence.

Step-by-Step Solution

Answer

Use \( S_n = \frac{n}{2} (a + l) \) or \( S_n = \frac{n}{2} (2a + (n-1)d) \), substituting the known terms and simplifying.

Step 1: Understand the Formula for the Sum

The sum of the first \( n \) terms of an arithmetic sequence can be calculated using the formula \[ S_n = \frac{n}{2} (a + l) \] where \( S_n \) is the sum of the first \( n \) terms, \( a \) is the first term, and \( l \) is the last term of the sequence. Alternatively, the formula \[ S_n = \frac{n}{2} (2a + (n-1)d) \] can be used, where \( d \) is the common difference between consecutive terms.

Step 2: Identify the Terms in the Sequence

To use either formula, identify the first term \( a \), the common difference \( d \), and the number of terms \( n \) in the sequence. If you are using the first formula, also identify the last term \( l \).

Step 3: Apply the Formula

Substitute the known values into the chosen formula. Calculate the expressions step by step, ensuring to simplify as needed to find the sum \( S_n \).

Step 4: Verify Your Calculation

Recheck each step of your calculation for errors to ensure accuracy. Confirm that the right formula was used with the correct substitutions.

Key Concepts

Sum of Arithmetic Sequence

The sum of an arithmetic sequence, often called an arithmetic series, is the total you get when you add up the numbers in a sequence that features a pattern of adding the same value – termed the common difference – to its terms. For the first few terms, let's imagine a sequence like this: 2, 4, 6, 8. Here, you simply add 2 each time to get the next number. Summing these numbers can be done using a handy formula. This formula is:
\[S_n = \frac{n}{2} (a + l)\]where:

• \( S_n \) is the sum of the sequence
• \( n \) is the number of terms
• \( a \) is the first term
• \( l \) is the last term

This formula is quite handy because it doesn’t require finding the middle terms – just the first and last ones. It’s often the easiest way, especially for students new to arithmetic sequences, to find the sum quickly.
Another alternative if the last term isn't known is given by:
\[S_n = \frac{n}{2} (2a + (n-1)d)\]This uses the common difference \( d \) to find the end of the sequence.

Common Difference

In an arithmetic sequence, consistency is key. The common difference, denoted by \( d \), is what makes that happen. It's the amount we continually add to each term to get the next one. Imagine we have a sequence like 3, 7, 11, 15. If we subtract 3 from 7, or 7 from 11, or 11 from 15, we always find \( d = 4 \). This regular interval is what distinguishes arithmetic sequences.
Calculating the common difference is straightforward:

• Choose any two consecutive terms
• Subtract the first term from the second
• That difference is \( d \)

Common difference is crucial for determining the sum of the sequence when the last term isn’t directly given, as it appears in the alternative sum formula.

Arithmetic Series Formula

The arithmetic series formula is a mathematical expression designed to help us quickly and easily find the sum of terms in an arithmetic sequence. It can take on two forms based on what information we may have available.
The first formula we use is:
\[S_n = \frac{n}{2} (a + l)\]used to calculate the series when we know the first and last terms.
The alternative formula is:
\[S_n = \frac{n}{2} (2a + (n-1)d)\]used when the last term isn't known, leveraging our knowledge of the sequence's common difference \( d \).
Both formulas require knowing:

• The number of terms \( n \)
• The first term \( a \)

This choice of formula depends on the context of the arithmetic sequence problem you’re solving, making it a flexible tool in mathematics.