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Chapter 5: Problem 39

Sigma notation Express the following sums using sigma notation. (Answers are not unique.) a. \(1+2+3+4+5\) b. \(4+5+6+7+8+9\) c. \(1^{2}+2^{2}+3^{2}+4^{2}\) d. \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\)

Short Answer

Expert verified
Question: Express the following sums using sigma notation: a. 1+2+3+4+5 b. 4+5+6+7+8+9 c. \(1^{2}+2^{2}+3^{2}+4^{2}\) d. 1+\(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\) Answer: a. \(\sum_{n=1}^{5} n\) b. \(\sum_{n=1}^{6} (n+3)\) c. \(\sum_{n=1}^{4} n^2\) d. \(\sum_{n=1}^{4} \frac{1}{n}\)

Step by step solution

01

Identify the formula for the terms in the sequence

We can see that each term of the sequence is a natural number that increases by 1 for each subsequent term. The general formula for this sequence is n, where n is the position of the term in the sequence.
02

Determine the starting and ending terms of the sequence

The first term of the sequence is 1, and the last term is 5.
03

Write the sum in sigma notation

Using the information we have gathered, we can write the sum in sigma notation as: \(\sum_{n=1}^{5} n\) b. \(4+5+6+7+8+9\)
04

Identify the formula for the terms in the sequence

Each term of the sequence is one more than the previous term. The general formula for this sequence is n + 3, where n is the position of the term in the sequence.
05

Determine the starting and ending terms of the sequence

The first term of the sequence is 4 (when n=1), and the last term is 9.
06

Write the sum in sigma notation

Using the information we have gathered, we can write the sum in sigma notation as: \(\sum_{n=1}^{6} (n+3)\) c. \(1^{2}+2^{2}+3^{2}+4^{2}\)
07

Identify the formula for the terms in the sequence

We can see that the sequence contains the square of each natural number. The general formula for this sequence is \(n^2\), where n is the position of the term in the sequence.
08

Determine the starting and ending terms of the sequence

The first term of the sequence is \(1^2\), and the last term is \(4^2\).
09

Write the sum in sigma notation

Using the information we have gathered, we can write the sum in sigma notation as: \(\sum_{n=1}^{4} n^2\) d. \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\)
10

Identify the formula for the terms in the sequence

The sequence contains the reciprocal of each natural number. The general formula for this sequence is \(\frac{1}{n}\), where n is the position of the term in the sequence.
11

Determine the starting and ending terms of the sequence

The first term of the sequence is 1 (when n=1), and the last term is \(\frac{1}{4}\) (when n=4).
12

Write the sum in sigma notation

Using the information we have gathered, we can write the sum in sigma notation as: \(\sum_{n=1}^{4} \frac{1}{n}\)

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