Problem 43 - Algebra for College Students

Question

Find the indicated sum. 1+5+9+13+\cdots+197 \quad 4950

Step-by-Step Solution

Answer

The sum of the sequence is 4950.

Step 1: Understanding the Sequence

We are asked to find the sum of the sequence 1 + 5 + 9 + 13 + ... + 197. This is an arithmetic sequence, where each term increases by a constant difference of 4. Our task is to determine the number of terms and then find their sum.

Step 2: Finding the Common Difference

Identify the common difference in the sequence, which is the difference between consecutive terms. By calculating 5 - 1, we find that the common difference (d) is 4.

Step 3: Calculating the Number of Terms

To find the number of terms (n), use the formula for the nth term of an arithmetic sequence: \( a_n = a_1 + (n-1) imes d \)Substitute the known values: 197 = 1 + (n-1) imes 4.Solve for n:\(197 = 1 + 4n - 4\)\(200 = 4n\)\(n = 50\). So, there are 50 terms in the sequence.

Step 4: Using the Sum Formula for Arithmetic Sequences

The sum of an arithmetic sequence can be found using the formula:\( S_n = \frac{n}{2} \times (a_1 + a_n) \)where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term. Substitute the values:\( S_{50} = \frac{50}{2} \times (1 + 197) \)\( S_{50} = 25 \times 198 \)\( S_{50} = 4950 \).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is the key factor that defines the sequence's pattern. This refers to the fixed amount that each subsequent term increases by in the sequence. For example, in the sequence 1, 5, 9, 13,..., the difference between each term is constant. To find this difference, you simply subtract any term from the following term. In our case:

• The second term minus the first term: 5 - 1 = 4

This tells us that the common difference, denoted as \(d\), is 4. Understanding the common difference is crucial because it helps you write the explicit formula for the sequence and predict future terms. Whenever you encounter an arithmetic sequence problem, identifying the common difference is one of the first steps. It essentially sets the rhythm of the sequence.

Sum of Arithmetic Sequence

When you need to find the sum of an arithmetic sequence, there is a helpful formula designed specifically for this purpose. This formula simplifies the calculation significantly:\[S_n = \frac{n}{2} \times (a_1 + a_n)\]Here's what each symbol represents:

• \(S_n\): The sum of the sequence
• \(n\): The number of terms in the sequence
• \(a_1\): The first term
• \(a_n\): The last term

To apply this formula to our exercise:

• We found \(n = 50\), the total number of terms.
• \(a_1 = 1\), the first term.
• \(a_n = 197\), the last term.

Plug these values into the formula:\[S_{50} = \frac{50}{2} \times (1 + 197)\]Simplifying this gives:

• \(S_{50} = 25 \times 198 = 4950\)

So, the sum of the sequence is 4950. Using this formula is very efficient, allowing you to skip adding up long lists of numbers.

Number of Terms in a Sequence

One critical aspect of working with an arithmetic sequence is determining the number of terms it contains. The good news is there's a straightforward formula for this, which revolves around the common difference (\(d\)) and the known terms in the sequence.The formula for the nth term of an arithmetic sequence is:\[a_n = a_1 + (n-1) \times d\]For our sequence, the nth term \(a_n\) is 197, the final term. We also know:

• \(a_1 = 1\), the first term.
• \(d = 4\), the common difference.

Putting these into the formula gives us:\[197 = 1 + (n-1) \times 4\]Solve this equation for \(n\):

• Subtract 1 from both sides: \(196 = 4n - 4\)
• Simplify: \(200 = 4n\)
• Divide by 4: \(n = 50\)

This tells us there are 50 terms in the sequence. Understanding how to find \(n\) helps you measure the span or length of the sequence, which is essential when calculating sums or predicting terms further along in the sequence.