Problem 61 - Precalculus: Mathematical for Calculus

Question

Write the sum using sigma notation. $$1^{2}+2^{2}+3^{2}+\dots+10^{2}$$

Step-by-Step Solution

Answer

The sum is written as \(\sum_{n=1}^{10} n^2\).

Step 1: Identify the Sequence

The sequence given is the squares of the first 10 natural numbers: \(1^2, 2^2, 3^2, \ldots, 10^2\).

Step 2: Determine the General Term

In this sequence, the general term for the \(n\)-th number is \(n^2\).

Step 3: Decide Limits of Summation

Since the sequence starts at 1 and ends at 10, our summation will start from \(n = 1\) and end at \(n = 10\).

Step 4: Write in Sigma Notation

Now, we can write the sum using sigma notation as follows:\[\sum_{n=1}^{10} n^2\]

Key Concepts

Summation

Summation is a mathematical process of adding sequences or series of numbers together. It's a fundamental concept used in algebra and calculus. To represent summation, we often use the Greek letter sigma (Σ), which stands for 'sum.'
In many math problems, summation allows you to express long, repetitive addition processes in a concise form. For example, instead of writing an addition of a series of numbers, you can use sigma notation briefly to communicate the same operation.

• **Sigma Notation**: The symbol for summation is Σ. It’s followed by an expression that incorporates a general term and its index.
• **Limits of Summation**: These are the values where the summation begins and ends. They are often represented at the bottom and top of the sigma: \sum_{\text{lower limit}}^{\text{upper limit}}

In the original exercise, the summation of squares of the first 10 natural numbers can be rewritten using sigma notation as \[\sum_{n=1}^{10} n^2 \]. This notation clearly shows the sum starts from 1 and ends at 10, simplifying the process of adding all these squares.

Sequences

A sequence is an ordered list of numbers that often follow a specific rule or pattern. Sequences are the building blocks of various mathematical concepts, including series and summations.
Different types of sequences include arithmetic sequences, geometric sequences, and quadratic sequences. Each type is distinguished by the manner in which its terms are generated from one another.

• **Arithmetic Sequence**: Where differences between consecutive terms are constant.
• **Geometric Sequence**: Where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
• **Quadratic Sequence**: Where the difference of differences between successive terms is constant.

In the problem, we dealt with the sequence of square numbers, which is a specific kind of quadratic sequence. The numbers \(1^2, 2^2, 3^2, \ldots, 10^2\) follow the pattern where each term is the square of its position in the sequence.

General Term

The general term of a sequence is a formula that gives the \(n\)-th term of a sequence. It allows you to find any term in the sequence without listing all the terms. Understanding the general term is essential for expressing sequences in sigma notation.

• **Purpose**: It provides a compact representation of a sequence, enabling calculations without exhaustive enumeration.
• **Notation**: Typically represented as \(a_n\) or in our case, \(n^2\), it conveys a rule based on the position \(n\).

In the example problem, the general term is \(n^2\), which means each term in the sequence is the square of its position in the list, \(n\). Knowing this allows you to replace any position in your sequence with its corresponding squared value, forming the basis for using sigma notation efficiently. This notation simplifies both the representation and the calculation to yield results using systematic steps. The term \(n^2\) is what is put into our sigma notation: \[\sum_{n=1}^{10} n^2 \].