Problem 13 - Calculus: One and Several Variables

Question

Express in sigma notation. $$1 \cdot 2+2 \cdot 3+3 \cdot 4+\dots-35 \cdot 36$$

Step-by-Step Solution

Answer

The given series can be expressed in the sigma notation as: \[\sum_{n = 1}^{35} n \cdot (n + 1)\]

Step 1: Identify the pattern in the series

The given series has the following structure: \(1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \dots + 35 \cdot 36\) We can observe that, for each term, the product of two consecutive integers is involved, and that the integers involved in the product are increasing. Let's try to find a general expression for the series terms.

Step 2: Derive a formula for the general term

We will use the variable \(n\) to represent the position of a term in the series. For example, \(n = 1\) for the first term, \(n = 2\) for the second term, and so on. In general, the term of the series at position \(n\) has the form: \(n \cdot (n + 1)\) Let's call this expression \(a_n\), which represents the \(n\)-th term of the series: \(a_n = n \cdot (n + 1)\)

Step 3: Express the series using sigma notation

Now that we have a formula for the general term, we can express the entire series using sigma notation. Since the series starts from the 1st term (corresponding to \(n = 1\)) and ends at the 35th term (corresponding to \(n = 35\)), the sigma notation is as follows: \(\sum_{n = 1}^{35} a_n = \sum_{n = 1}^{35} n \cdot (n + 1)\) The given series can be expressed in the sigma notation as: \[\sum_{n = 1}^{35} n \cdot (n + 1)\]

Key Concepts

Series and Sequences

Understanding **series and sequences** is crucial in mathematics, especially in calculus and algebra. In simple terms, a sequence is a list of numbers following a certain pattern. A series, on the other hand, is the sum of the terms of a sequence.

For example, consider the sequence of numbers: 1, 2, 3, and so on. If we sum these numbers, we form a series.

In this exercise, the pattern in the sequence involves the product of consecutive integers. Recognizing the pattern allows us to apply formulas and tools, like sigma notation, to work with the series efficiently.

• **Sequence Example:** 1, 2, 3, 4...
• **Series Example:** 1 + 2 + 3 + 4...

Recognizing these structures helps solve problems involving large numbers of terms efficiently.

Consecutive Integers

**Consecutive integers** are integers that follow each other in order. They differ by one. You can think of them as numbers in a row; for example, 1, 2, 3 are consecutive integers.

When you look at the series in the exercise, each term is formed by multiplying two consecutive integers. Specifically, the number and the next number in the sequence.

• The first term: 1 and 2
• The second term: 2 and 3
• The third term: 3 and 4

This way of representing numbers makes it easy to recognize patterns and can be expressed using general formulas, like in sigma notation. Understanding this concept is helpful for simplifying and solving sequences in advanced mathematics.

Mathematical Notation

**Mathematical notation** is a language used by mathematicians to write formulas and equations concisely. Sigma notation, in particular, is a method for expressing a series. It uses the Greek letter \(\Sigma\) to represent the sum of a sequence of terms.

The expression \(\sum_{n=1}^{35} n \cdot (n+1)\), from the exercise, is an example of sigma notation. Here's what it means:

• **\(\Sigma\)**: Signifies summation.
• **\(n=1\) to \(n=35\)**: Indicates that the series starts with \(n=1\) and ends with \(n=35\).
• **\(n \cdot (n+1)\)**: Represents each term in the series, showing the multiplication of consecutive integers.

By using sigma notation, complex series can be condensed into a single expression, making it easier to manipulate and understand. This shorthand is invaluable for addressing lengthy calculations efficiently.