Problem 15 - Essential Calculus

Question

In Exercises 13-16, use the properties of summation and Theorem 4.2 to evaluate the sum. Use the summation capabilities of a graphing utility to verify your result. $$ \sum_{i=1}^{20}(i-1)^{2} $$

Step-by-Step Solution

Answer

The sum \(\sum_{i=1}^{20}(i-1)^{2}\) equals 2470.

Step 1: Shift the sequence

The summation term is \((i-1)^{2}\) summing from \(i=1\) to \(i=20\). Notice that you can rewrite this sum to start from 0 instead of 1, because \((i-1)^{2}\) for \(i=1\) is essentially 0. So, the sum can be rewritten as \(\sum_{i=0}^{19}(i)^{2}\)

Step 2: Apply the sum of squares formula

Using Theorem 4.2, the sum of squares of the first n natural numbers is \(n*(n+1)*(2n+1)/6\). Here \(n=19\), so substituting this into the formula gives \(19*20*39/6\).

Step 3: Simplify the expression

The product \(19*20*39/6\) simplifies to 2470

Key Concepts

Sigma Notation

Sigma notation, represented by the Greek letter \( \Sigma \), is a mathematical convention used to denote the sum of a sequence of terms. This efficient symbol allows mathematicians and students alike to express the addition of a series of numbers or expressions in a compact form. It significantly reduces complexity and potential for error when dealing with long sequences.

For instance, the sum \( \sum_{i=1}^{n} a_i \) tells us to add all terms \( a_i \) from \( i=1 \) to \( i=n \). Each \( a_i \) is a term in the sequence, and \( i \) is the index that indicates the position of each term within the sequence. By understanding sigma notation, students can more easily work with series and mathematical proofs and interpret the sum of complex sequences.

Sum of Squares Formula

One valuable formula often used in conjunction with sigma notation is the sum of squares formula. This formula is a special case that allows for the calculation of the sum of the squares of the first \( n \) natural numbers. The formula is expressed as \[ \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} \].

Using this formula saves time and computational effort, particularly when dealing with large values of \( n \). It is derived from a more general approach to calculating sums of powers and plays a crucial role in statistical formulas, such as the calculation of variance. Moreover, the sum of squares formula is an excellent example of how sequences can be handled algebraically rather than by brute force calculation.

Graphing Utility Verification

Graphing utilities, such as calculators or computer software, can be extremely helpful in verifying the results of sums, especially when dealing with complicated expressions. After calculating the sum analytically using formulas, properties, and manipulations, plugging the initial problem or the equivalent manipulated expression into a graphing utility can confirm whether the analytical work was correct.

To verify a sum like \( \sum_{i=1}^{n} f(i) \) using a graphing utility, one would typically enter the sum into the utility using available summation functions. Correctly configuring the range and expression will yield the sum, which can then be compared to the manually derived result. This step acts as a quality check and enhances understanding by providing a visual representation of the summands and sum.

Sequence Shifting

Sequence shifting is a technique often utilized in algebra to transform the index of summation to a more convenient starting point. In sigma notation, it is sometimes helpful to adjust the lower and upper bounds of summation to simplify the terms being summed. This step is particularly useful if it aligns with a known formula or property.

For example, shifting the sequence from \( \sum_{i=1}^{20}(i-1)^{2} \) to \( \sum_{i=0}^{19}i^2 \) simplifies the expression, making it more amenable to direct application of formulas, like the sum of squares formula. Sequence shifting can be a powerful tool in solving summation problems effectively. However, it is essential to ensure that such a shift does not alter the value of the sum, as only a rearrangement of terms is permitted, not omission or addition.