Problem 47 - College Algebra

Question

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$1+2+3+\dots+30$$

Step-by-Step Solution

Answer

The sum \(1+2+3+\ldots+30\) can be expressed in summation notation as \(\Sigma_{{i=1}}^{30} i\).

Step 1: Identify the series type and pattern

The given series is a list of consecutive natural numbers from 1 to 30. The series follows an arithmetic pattern where each succeeding number is increased by 1.

Step 2: Define the General Form of Series

In an arithmetic series, each term can be defined by a formula. Since this series starts at 1 and increments by 1, we can say the ith term, denoted by \(a_i\), is exactly \(i\). So we can express any term in the series as \(a_i = i\).

Step 3: Translating to Summation Notation

Sigma notation, also known as summation notation, consists of three parts. It has an uppercase sigma (\(\Sigma\)), a number at the bottom called the lower limit of summation (in this case, 1), and a number at the top called the upper limit of summation (in this case, 30). In the middle, we place the general form of the series \(a_i = i\). Putting all this together we write the series in the summation notation as \(\Sigma_{{i=1}}^{30} i\)