In Exercises 10–11, express each sum using summation notation. Use i for the index of summation. 1/3 + 2/4 + 3/5 + ... + 15/17

Verified step by step guidance

1. Identify the pattern in the sequence: The numerators are consecutive integers starting from 1, and the denominators are consecutive integers starting from 3.
2. Express the general term of the sequence: The nth term can be expressed as \( \frac{n}{n+2} \).
3. Determine the range of the index of summation: The sequence starts with 1/3 and ends with 15/17, so the index i starts at 1 and ends at 15.
4. Write the sum in summation notation: Use the general term and the range of the index to express the sum as \( \sum_{i=1}^{15} \frac{i}{i+2} \).
5. Verify the expression: Ensure that the summation notation correctly represents the given sequence by checking the first and last terms.

Key Concepts

Summation Notation

Summation notation is a mathematical shorthand used to represent the sum of a sequence of terms. It typically uses the Greek letter sigma (Σ) to denote the sum, along with an index of summation that indicates the starting and ending values. For example, Σ from i=1 to n of a_i represents the sum of the terms a_1, a_2, ..., a_n.

Index of Summation

The index of summation is a variable that represents the position of each term in the sequence being summed. It is usually denoted by a letter, commonly 'i', and it takes on integer values starting from a specified lower limit to an upper limit. Understanding how to manipulate the index is crucial for correctly expressing sums in summation notation.

Pattern Recognition in Sequences

Pattern recognition in sequences involves identifying a consistent rule or formula that describes the terms of the sequence. In the given sum, recognizing that the numerator increases by 1 and the denominator increases by 1 as well helps in formulating the general term. This ability to discern patterns is essential for accurately expressing sums in summation notation.