Problem 7 - Algebra for College Students

Question

Find the sum of each series. $$\sum_{i=1}^{4} i^{2}$$

Step-by-Step Solution

Answer

The sum of the series is 30.

Step 1 - Write down the series

The given series is \(\textstyle\sum_{i=1}^{4} i^{2}\). This means we need to find the sum of squares of the first four positive integers.

Step 2 - List the terms

List each term of the series: \(i = 1^{2}, 2^{2}, 3^{2}, 4^{2}\).

Step 3 - Calculate each term

Calculate each squared value: \(1^{2} = 1, 2^{2} = 4, 3^{2} = 9, 4^{2} = 16\).

Step 4 - Sum the terms

Add the squared values together: \(1 + 4 + 9 + 16\) which equals \(30\).

Key Concepts

summing squares

When we talk about 'summing squares,' we mean finding the sum of the squares of a sequence of numbers. In this exercise, the sequence is the first four positive integers.
For instance, the series given is \(\textstyle\sum_{i=1}^{4} i^{2} \), which asks us to sum the squares of numbers from 1 to 4.
To sum the squares:

• Write down each number.
• Square each number.
• Sum these squared numbers together.

This is a common method used in various areas of math to simplify complex expressions and can often be seen in statistical calculations, physics, and engineering.

series calculation

Understanding 'series calculation' is essential for adding sequences of numbers. A series is simply a sum of terms in a sequence. In this case, our series is formed by the squares of the first four positive integers.
The general approach for series calculation involves:

• Identifying the type of series (e.g., arithmetic, geometric, or others).
• Listing the terms of the series.
• Calculating individual terms if needed.
• Summing these terms together.

Here, we had to specifically find the sum of squares. We listed and calculated each term as:
\(\textstyle 1^{2} = 1 \), \(\textstyle 2^{2} = 4 \), \(\textstyle 3^{2} = 9 \), \(\textstyle 4^{2} = 16 \).
Summing these gives us the final result: \textstyle 1 + 4 + 9 + 16 = 30 \.

basic algebra

Basic algebra is fundamental to solving problems like the one given, which involves summing squares. In algebra, you deal with symbols and the rules for manipulating these symbols to solve equations or find sums.
To handle this exercise with basic algebra, follow these steps:

• Recognize the series format \(\textstyle \sum_{i=1}^{4} i^{2} \).
• Identify each element in the sequence based on the given range.
• Square each element using the algebraic rule \(\textstyle i^{2} \).
• Add up all resulting squares to get the final sum.

In practice, mastering basic algebra enables you to break down complex problems into manageable parts, just like we did with each step in our series calculation. This clarity in steps ensures you can replicate the process for other series calculations as well.