Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=n^{2} $$

Step-by-Step Solution

Answer

The first five terms are 1, 4, 9, 16, and 25.

Step 1: Understand the Sequence

The sequence is defined by the general term \(a_n = n^2\). This means for each term \(a_n\), you square the position \(n\) in the sequence.

Step 2: Calculate the First Term

The first term \(a_1\) is calculated by substituting \(n = 1\) into the formula \(a_n = n^2\). Thus, \(a_1 = 1^2 = 1\).

Step 3: Calculate the Second Term

To find the second term \(a_2\), substitute \(n = 2\) into the formula to get \(a_2 = 2^2 = 4\).

Step 4: Calculate the Third Term

For the third term, substitute \(n = 3\) into the formula: \(a_3 = 3^2 = 9\).

Step 5: Calculate the Fourth Term

The fourth term \(a_4\) is found by substituting \(n = 4\) into the formula: \(a_4 = 4^2 = 16\).

Step 6: Calculate the Fifth Term

Finally, for the fifth term, substitute \(n = 5\) into the formula to get \(a_5 = 5^2 = 25\).

Key Concepts

Algebraic Expressions

Algebraic expressions are like a secret code that describe mathematical worlds with letters and numbers; they are central in mathematics. Imagine them as sentences that tell us what operation to perform. In our exercise, the expression \( a_n = n^2 \) tells us exactly how to construct each term in the sequence.

• **Variables**: In this expression, "\( n \)" is a variable. Variables act like placeholders. They can take on different values just like how pronouns can represent different characters in a story.
• **Operations**: Here, \( n^2 \) means the variable \( n \) is squared—that is, multiplied by itself.
• **Constants**: Though not present in this specific expression, constants are numbers that don't change value, such as \(5\) in \( x + 5 \).

To understand any algebraic expression, you need to decipher what each letter or symbol represents and what mathematical operations are being instructed. It's like learning the rules of a game so you can play it right!

Square Numbers

Square numbers are the result of multiplying an integer by itself. They appear frequently in both nature and equations. Knowing them can help you recognize patterns and solve problems faster.

• A square number arises by multiplying a number by itself. For example, \(4\), because it is \(2\times 2\), and so on.
• If you see \( n^2 \), it invites you into a world where simple multiplication breathes new life into numbers. \( n=3 \) yields a square number of \( 9 \).

Square numbers have some fascinating properties:

• They are always positive.
• They grow quickly as their base number increases.
• They can be visually represented by squares arranged in a grid.

Grasping square numbers helps build a solid math foundation. They are stepping stones in your mathematical journey, easing the way through more complex arithmetic and algebra.

General Term of a Sequence

The general term of a sequence is an algebraic formula that allows us to find any term in the sequence, without listing all preceding terms. It's your mathematical GPS, guiding you directly to the term you need.Our exercise gives us the sequence \( a_n = n^2 \). With this formula:

• You can find the first term by replacing \( n \) with 1, \( a_1 = 1^2 = 1 \).
• To find the second term, use \( n = 2 \), resulting in \( a_2 = 4 \).
• The same goes for the third, fourth, or nth term.

This formula holds power:

• It saves time compared to calculating each square individually.
• It unveils the underlying pattern quickly.
• It makes predictions possible about unknown terms.

Understanding how to utilize a general term in a sequence equips you to explore sequences of all kinds efficiently, and is a crucial skill in both basic and advanced mathematics.