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. Step 1: Identify the general term of the sequence. Observe the pattern in the numerators and denominators of the fractions: the numerator increases by 1 starting from 1, and the denominator increases by 1 starting from 3. The general term can be expressed as $\frac{i}{i}$, where i is the index of summation.
2. Step 2: Determine the range of the index of summation. The sequence starts with the fraction $\frac{1}{3}$ (when i = 1) and ends with the fraction $\frac{15}{17}$ (when i = 15). Therefore, the index i ranges from 1 to 15.
3. Step 3: Write the summation notation. Combine the general term and the range of the index to express the sum in summation notation: $\sum i_{1}15\frac{i}{i}$.
4. Step 4: Verify the summation notation. Check that the general term and the range of the index correctly represent the given sequence. Substitute a few values of i (e.g., i = 1, i = 2, i = 15) into the summation notation to confirm that it matches the original sequence.
5. Step 5: Finalize the summation notation. The sum can now be expressed as $\sum i_{1}15\frac{i}{i}$, which represents the sequence 1/3 + 2/4 + 3/5 + ... + 15/17.

Key Concepts

Summation Notation

Summation notation is a mathematical shorthand used to represent the sum of a sequence of terms. It is typically denoted by the Greek letter sigma (Σ), followed by an expression that defines the terms to be summed, along with limits that specify the starting and ending indices. For example, Σ from i=1 to n of a_i indicates 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. In the expression Σ from i=a to b of f(i), 'i' is the index that takes on integer values from 'a' to 'b'. This index allows us to systematically enumerate each term in the sum, making it easier to express complex series in a concise form.

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. In the given sum, the numerators (1, 2, 3, ..., 15) form a simple arithmetic sequence where each term increases by 1. Understanding the properties of arithmetic sequences is essential for identifying patterns and expressing sums in summation notation.