Question

Express each sum using summation notation. \(1+3+5+7+\cdots+[2(12)-1]\)

Short answer

\( \sum_{n=1}^{12} (2n-1) \)

Step-by-step solution

- Identify the sequence pattern

The given sequence is 1, 3, 5, 7, ..., which is an arithmetic sequence where each term increases by 2. This can be represented as the sequence of odd numbers.

- Express the general term

For an arithmetic sequence starting at 1 with a common difference of 2, the nth term can be expressed as: \[ a_n = 1 + (n-1) \times 2 = 2n - 1 \]

- Determine the number of terms

We need to find the value of the last term in the series. The last term is given as \[ 2(12) - 1 = 23 \]. To express 23 in terms of our general formula, set \[ 2n - 1 = 23 \]. Solving for n, we get: \[ 2n = 24 \] \[ n = 12 \] Thus, there are 12 terms in the series.

- Write the summation notation

Now express the sum using summation notation from the first term to the 12th term. The summation notation for the series is: \[ \sum_{n=1}^{12} (2n-1) \]

Key concepts

arithmetic sequence

In an arithmetic sequence, each term is generated by adding a constant value, known as the common difference, to the previous term. Let's look at the sequence 1, 3, 5, 7,... from the exercise. Each number increases by 2, so the common difference is 2. This sequence can be described as the list of odd numbers because the difference between each term is always the same. Recognizing this pattern helps in forming a general rule to find any term in the sequence.

general term

To find any term in an arithmetic sequence, use the general term formula. For the sequence starting at 1 with a common difference of 2, the general term is given by:
\[ a_n = a_1 + (n-1) \times d \]
where \(a_1\) is the first term and \(d\) is the common difference. In our case, \(a_1 = 1\) and \(d = 2\), so the formula becomes:
\[ a_n = 1 + (n-1) \times 2 = 2n - 1 \]
This formula allows you to calculate the nth term. For example, if you want the 5th term, substitute n with 5 to get:
\( a_5 = 2(5) - 1 = 9 \)

series sum

When dealing with a series, we are interested in summing all terms from the first to the nth term. The given series is 1 + 3 + 5 + 7 + ... + [2(12)-1]. To sum this, we'll use summation notation:
\[ \sum_{n=1}^{12} (2n - 1) \]
This notation makes it easy to write out and calculate the sum of the series. By recognizing the pattern and using the general term, we can simplify and find the sum without writing every single term.

sequence pattern

Identifying patterns in a sequence is crucial. In our example, the pattern is that every number is an odd number, which can be generalized as \(2n - 1\). Once we recognize this, solving for the number of terms becomes straightforward. For the last term 23, we set up an equation:
\(2n - 1 = 23 \)
Solving gives us \(n = 12\), confirming there are 12 terms. This step-by-step method of identifying the pattern, finding the general term, solving for the number of terms, and then summing, simplifies understanding complex sequences.