Problem 18 - Algebra 2 and Trigonometry

Question

In \(9-18,\) use the given information to a. write the series in sigma notation, and b. find the sum of the first \(n\) terms. $$ a_{5}=15, d=2, n=12 $$

Step-by-Step Solution

Answer

Series in sigma notation: \( \sum_{i=1}^{12} (7 + (i-1) \times 2) \); Sum: 216.

Step 1 - Identify Key Information

We are given the information \( a_5 = 15 \), \( d = 2 \), and \( n = 12 \). This means we have an arithmetic sequence where the 5th term is 15, the common difference \( d \) is 2, and we need to find the first 12 terms.

Step 2 - Find the First Term

The 5th term of an arithmetic sequence can be found using the formula \( a_5 = a + 4d \). We know \( a_5 = 15 \) and \( d = 2 \), so we substitute these values into the equation: \( 15 = a + 4(2) \). This simplifies to \( 15 = a + 8 \), giving us \( a = 7 \).

Step 3 - Write the Series in Sigma Notation

An arithmetic sequence can be expressed as a series in sigma notation. For our sequence, the general term is \( a_n = a + (n-1)d \). Since \( a = 7 \) and \( d = 2 \), the general term becomes \( a_n = 7 + (n-1)(2) \). Thus, the series can be written in sigma notation as: \[ \sum_{i=1}^{12} (7 + (i-1) \, \times \, 2) \]

Step 4 - Calculate the Sum of the Series

The sum of the first \( n \) terms of an arithmetic series is given by the formula \( S_n = \frac{n}{2} (2a + (n-1)d) \). Substitute \( n = 12 \), \( a = 7 \), and \( d = 2 \): \[ S_{12} = \frac{12}{2} (2 \times 7 + (12-1) \times 2) \] Simplify this to calculate the sum: \( S_{12} = 6 (14 + 22) = 6 \times 36 = 216 \). Thus, the sum of the first 12 terms is 216.

Key Concepts

Sigma Notation

Sigma notation is a mathematical way to express the sum of a sequence using the Greek letter sigma (a3). It's a way to compactly write out the series, which can otherwise be cumbersome if written in full. For instance, if we want to sum up terms from an arithmetic sequence, we can use sigma notation to represent this sum.
For our exercise, we have an arithmetic sequence where the first term is 7, the common difference is 2, and we are summing the first 12 terms. In sigma notation, this looks like:

• a3 (from i=1 to 12) of (7 + (i-1) a00)

This captures the start of the sequence, the number of terms, and the pattern of each term in compact form. It tells us that we're adding up terms as they progress from the first term to the twelfth.

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is obtained by adding a constant value to the previous term. This constant is known as the "common difference."
In mathematical terms, an arithmetic sequence is expressed as:

• First term: \( a \)
• Second term: \( a + d \)
• Third term: \( a + 2d \)
• And so on...

In our example, we determined that the first term \( a = 7 \) and the common difference \( d = 2 \). So, the sequence begins with 7, then 9, 11, and continues by adding 2 to each subsequent term until we reach 12 terms.

Sum of Series

The sum of an arithmetic series is the total of all terms in the sequence up to a specific point. Finding this sum can be easily done using a formula rather than adding each term one by one.
To find the sum of the first \( n \) terms of an arithmetic sequence, we use the formula:

• \( S_n = \frac{n}{2} \times (2a + (n-1)d) \)

In our example, to find the sum of the first 12 terms, we substitute the known values: \( a = 7 \), \( d = 2 \), and \( n = 12 \).
The sum is \( S_{12} = \frac{12}{2} \times (14 + 22) = 6 \times 36 = 216 \).
This shows the brevity of using the sum formula, allowing us to quickly compute without manually adding each term in the series.

Common Difference

The common difference in an arithmetic sequence is the constant amount that each term increases or decreases from the previous term. It determines the "step" by which the sequence progresses.
For an arithmetic sequence:

• The second term minus the first term gives the common difference.
• Similar calculations apply for other terms.

In our specific problem, the common difference is \( d = 2 \). This means that beginning with our initial term of 7, each subsequent term is 2 more than the previous one. Calculating this difference consistently confirms that the sequence is arithmetic.
Understanding this step size helps us construct and interpret the series effectively, ensuring the pattern holds true throughout all terms.