Problem 89 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the sum. $$\sum_{i=0}^{6} 4 i^{2}$$

Step-by-Step Solution

Answer

The sum is 364

Step 1: Identify the Pattern and Summation Formula

The given problem is a series of the form \(4i^{2}\) where \(i\) ranges from 0 to 6. This is a simple arithmetic progression of squares scaled by a constant factor of 4. We can rewrite this series using the summation which is represented by the Greek symbol sigma \(\sum\). For any integer \(n\), the sum of the squares of the first \(n\) positive integers is given by the formula \(\frac{n(n+1)(2n+1)}{6}\). This means we'll need to substitute 6 (the given upper limit) in this formula to get our answer.

Step 2: Apply the Summation Formula

We have the expression \(\sum_{i=0}^{6} 4 i^{2}\), which can also be written as \(4 \sum_{i=0}^{6} i^{2}\). Applying the formula \(\frac{n(n+1)(2n+1)}{6}\) where \(n = 6\) we get \(4 * \frac{6 * (6 + 1) * (2*6 + 1)}{6}\)

Step 3: Compute the Result

After doing the arithmetic, we end up with \(4 * \frac{6 * 7 * 13}{6}\). Simplify this expression to get the final result

Key Concepts

Arithmetic Progression

Arithmetic progression, often abbreviated as AP, is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the preceding term. This constant difference is known as the common difference.

For example, in the sequence 2, 5, 8, 11, the common difference is 3. If everyone understands how arithmetic progressions work, tackling sequences and series becomes much more straightforward.

Even though the problem originally mentioned involves squares (making it a bit more complex than a linear AP), understanding basic arithmetic progression concepts lays the groundwork for understanding more complicated sequences.

Sum of Squares

The sum of squares is a concept where each term in a sequence is squared, and these squares are summed up. It's important in various branches of mathematics, including algebra and statistics.

• The general formula to find the sum of squares of the first n integers is \( \frac{n(n+1)(2n+1)}{6} \).
• This formula helps condense calculations instead of adding each square individually, which is beneficial for large n.

In our problem, we use this formula by setting \( n \) to 6. This lets us find the sum of squares for integers from 0 to 6 easily.

Sigma Notation

Sigma notation is a concise way of expressing the sum of a set of terms. The Greek letter \( \Sigma \) is used to denote that a sequence of elements is summed up.

• The expression \( \sum_{i=0}^{6} 4i^{2} \) indicates that we're summing 4 times the square of each integer from 0 to 6.
• It provides a shorthand way of writing out repeated addition, especially in cases involving mathematical series.

Utilizing sigma notation helps in recognizing patterns and applying formulas to solve the problem efficiently.

Integer Sequence

An integer sequence is a list of whole numbers ordered in a specific way. Each number in the sequence is called a term.

These sequences can follow various patterns, like arithmetic or geometric, where each term depends on its position or previous terms.

In the original problem, the sequence we deal with is created by squaring each integer from 0 to 6 and then multiplying by 4. Understanding the underlying pattern helps us choose the correct formulas and techniques to find the sum quickly.

Grasping integer sequences builds foundational mathematical skills applicable in advanced studies like algebra and calculus.