In Exercises 55–60, express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. 5+7+9+11+⋯+ 31

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Identify the pattern in the series: 5, 7, 9, 11, ..., 31. Notice that each term increases by 2, so this is an arithmetic sequence with first term \(a_1 = 5\) and common difference \(d = 2\).

Express the general term of the sequence using the formula for the \(k\)-th term of an arithmetic sequence: \(a_k = a_1 + (k - 1)d\). Substitute \(a_1 = 5\) and \(d = 2\) to get \(a_k = 5 + 2(k - 1)\).

Determine the number of terms in the sequence by setting the general term equal to the last term: \(5 + 2(k - 1) = 31\). Solve this equation for \(k\) to find the upper limit of summation.

Choose the lower limit of summation as \(k = 1\) (since the first term corresponds to \(k=1\)), and the upper limit as the value of \(k\) found in the previous step.

Write the sum in summation notation as \(\sum_{k=1}^{n} \left(5 + 2(k - 1)\right)\), where \(n\) is the number of terms found. This expresses the entire sum using summation notation.

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주요 개념

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Arithmetic Sequences

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. In this problem, the sequence 5, 7, 9, 11, ..., 31 increases by 2 each time, which helps identify the general term formula needed for summation notation.

Summation Notation (Sigma Notation)

Summation notation uses the Greek letter sigma (∑) to represent the sum of terms in a sequence. It includes an index of summation, lower and upper limits, and a general term expression, allowing a compact representation of long sums.

General Term of an Arithmetic Sequence

The general term of an arithmetic sequence is given by a formula like a_k = a_1 + (k - 1)d, where a_1 is the first term and d is the common difference. This formula is essential to express each term in the sum as a function of k for summation notation.