Problem 50 - College Algebra

Question

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$\frac{1}{3}+\frac{2}{4}+\frac{3}{5}+\dots+\frac{16}{16+2}$$

Step-by-Step Solution

Answer

The sum is expressed in summation notation as \( \sum_{i=1}^{16} \frac{i}{i+2} \)

Step 1: Identify pattern

Look at the pattern in the sequence. Notice that the sequence starts from 1/3 and increases by one in the numerator and denominator for each subsequent term. This suggests that the index of summation, i, is involved in each term in both the numerator and denominator.

Step 2: Express sequence in terms of i

Express each term of the sequence in terms of i. Seeing as the numerator of each fraction is the same as the value of i (starting at 1), the numerator can be expressed as i. The denominator is always the corresponding numerator (i) plus 2, and therefore can be expressed as (i+2).

Step 3: Write summation notation

Write the expression in summation notation starting at i=1 as the problem states. This means expressing the sequence as the sum from i=1 to i=16 of i/(i+2)