Question

Find the value of the indicated sum. $$ \sum_{i=1}^{6} i^{2} $$

Short answer

The sum is 91.

Step-by-step solution

Understand the Summation Notation

The expression \( \sum_{i=1}^{6} i^2 \) means that we need to find the sum of the squares of integers from 1 to 6. Specifically, we need to calculate \( 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 \).

Calculate Each Individual Square

Compute the square of each integer from 1 to 6: \(1^2 = 1\), \(2^2 = 4\), \(3^2 = 9\), \(4^2 = 16\), \(5^2 = 25\), \(6^2 = 36\).

Sum the Squares

Add up all the calculated squares: \[ 1 + 4 + 9 + 16 + 25 + 36 = 91 \]

Verify the Calculation

Double-check each square and ensure their sum is correct. Re-calculate if necessary to confirm the total is \(91\).

Key concepts

Definite Sum

In mathematics, a definite sum refers to the sum of a sequence of terms within a fixed range. This concept is integral when using summation notation, symbolized by the Greek letter sigma (\( \Sigma \)). The expression \( \sum_{i=1}^{6} i^2 \) is a classic example of a definite sum. Here, \( i \) serves as an index that runs from a starting point (1 in this case) to an endpoint (6). Each value of \( i \) is used to compute subsequent terms in the sequence or series.

Understanding definite sums is crucial as they allow us to calculate totals of sequences efficiently. This process involves adding up individual computed values that follow a particular set pattern or rule. In real-world terms, think of definite sums as being similar to carefully counting items on a list, ensuring that you don’t miss any or add extras.

For example, if you want to sum the integers' squared values from 1 to 6 using the notation \( \sum_{i=1}^{6} i^2 \), you will compute as follows: \( 1^2, 2^2, 3^2 \ldots \) up to \( 6^2 \). When each square is computed and summed from the first integer to the sixth, we achieve the result inside a specified range.

Square of Integers

The term 'square of integers' refers to multiplying an integer by itself, such as \( i^2 \). Each integer's square represents a form of exponential growth where the number is raised to the power of two. For example, squaring the number 2 gives you \( 4 \) because \( 2 \times 2 = 4 \).

In the context of the exercise, the series involves squaring each integer from 1 to 6. This results in a set of squared values: \( 1^2 = 1 \), \( 2^2 = 4 \), \( 3^2 = 9 \), \( 4^2 = 16 \), \( 5^2 = 25 \), \( 6^2 = 36 \). Observing these numbers reveals an interesting pattern of growth, as the squares increase with the increase in base integer numbers.

Understanding how to calculate and interpret the squares of integers is fundamental in many areas of math, such as algebra and calculus. It is also just as important in practical scenarios, where geometric calculations or solving quadratic equations might require these basic computations.

Series Calculation

A series is the sum of the terms of a sequence, and series calculation is a method used to determine this sum. In mathematical terms, a sequence is an ordered list of numbers, while a series is formed when these numbers are added together. The specific sum involving the square of integers \( \sum_{i=1}^{6} i^2 \) exemplifies this concept by merging individual computations into a single total.

To calculate the series, you first determine each term individually—like the squares found earlier—and then proceed to sum these results. For instance, adding numbers like \( 1 + 4 + 9 + 16 + 25 + 36 \) provides the full series sum of \( 91 \).

Series calculations are invaluable as they arise frequently in various mathematical and real-life applications, from financial calculations such as interest rates over time to physical principles like calculating distance or energy. Understanding how to execute a series calculation is essential for tackling more advanced topics and problems, offering a deeper comprehension of how individual components come together to form an entire whole.