Problem 11 - College Algebra

Question

Express each arithmetic sum using summation notation. $$ 10+18+26+\ldots+162 $$

Step-by-Step Solution

Answer

The sum is expressed as \( \sum_{n=1}^{20} (8n + 2) \).

Step 1: Identify the sequence type

The sequence given is 10, 18, 26,..., 162. Each term increases by a constant difference. Therefore, it is an arithmetic sequence.

Step 2: Determine the common difference

Subtract the first term from the second term: 18 - 10 = 8. So, the common difference \( d = 8 \).

Step 3: Find the first term

The first term of the sequence is \( a_1 = 10 \).

Step 4: Determine the general term of the sequence

The general formula for the nth term of an arithmetic sequence is given by \( a_n = a_1 + (n-1)d \). Substituting the known values: \( a_n = 10 + (n-1) \times 8 = 10 + 8n - 8 = 8n + 2 \).

Step 5: Find n when a_n = 162

Set \( a_n = 162 \) and solve for \( n \). \[ 162 = 8n + 2 \] Subtract 2 from both sides: \[ 160 = 8n \] Divide both sides by 8:\[ n = 20 \] So, there are 20 terms in the sequence.

Step 6: Express the sum in summation notation

The sum of the first n terms of an arithmetic sequence can be expressed in summation notation. The sum \( S \) is \[ S = \sum_{n=1}^{20} (8n + 2) \]

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each term after the first is derived by adding a constant value, known as the common difference, to the preceding term. This means that if you know the first term of an arithmetic sequence, you can find any other term by repeatedly adding the common difference.

For example, the sequence 10, 18, 26, ..., 162 is an arithmetic sequence because each term is formed by adding 8 to the previous term. Notice that this creates a very predictable pattern, which makes arithmetic sequences particularly useful for calculations involving many numbers. They simplify computations since you can express any number in the sequence using simple arithmetic.

Common Difference

The common difference is the fixed number you add to each term of an arithmetic sequence to get the next term. It can also be negative, which results in a decreasing sequence. In our example, the common difference is determined by subtracting the first term from the second term, i.e.,

• Second term: 18
• First term: 10
• Common difference, \(d = 18 - 10 = 8\)

Understanding the common difference allows you to generate successive terms of the sequence and also helps in constructing a general formula for any term in the sequence. This knowledge lets one jump between different terms of the sequence with ease.

General Term of a Sequence

The general term of an arithmetic sequence describes any term in the sequence using the position of the term. The general term is often expressed as \(a_n = a_1 + (n-1)d\), where:

• \(a_n\) is the nth term.
• \(a_1\) is the first term.
• \(d\) is the common difference.
• \(n\) is the term number.

In the example sequence 10, 18, 26, ..., 162, the first term \(a_1 = 10\) and the common difference \(d = 8\). Plugging these into the formula gives:\[ a_n = 10 + (n-1) \times 8 = 8n + 2 \]This formula allows you to calculate any term in the sequence without listing all prior terms, making it particularly useful for larger sequences.

Sum of Terms

To find the sum of the terms in an arithmetic sequence, you can use summation notation. This involves adding up all terms in the sequence from the first to the nth term. The sum of an arithmetic sequence can be expressed as:\[ S_n = \sum_{i=1}^{n} a_i \]For our example, the sequence is represented as \( 8n + 2\) from \(n = 1\) to \(n = 20\) which gives:\[ S = \sum_{n=1}^{20} (8n + 2) \]Understanding and utilizing summation notation allows for elegant expression of sums, especially for a large number of terms, without explicit computations of each individual term.