Problem 9 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

The first five terms are 3, 9, 27, 81, and 243.

Step 1: Identify the sequence rule

The formula given for the sequence is \(a_{n} = 3^{n}\). This means each term \(a_{n}\) is calculated by using \(3\) raised to the power of \(n\).

Step 2: Calculate the first term

To find the first term \(a_1\), substitute \(n = 1\) into the formula: \[a_1 = 3^1 = 3\].

Step 3: Calculate the second term

To find the second term \(a_2\), substitute \(n = 2\) into the formula: \[a_2 = 3^2 = 9\].

Step 4: Calculate the third term

To find the third term \(a_3\), substitute \(n = 3\) into the formula: \[a_3 = 3^3 = 27\].

Step 5: Calculate the fourth term

To find the fourth term \(a_4\), substitute \(n = 4\) into the formula: \[a_4 = 3^4 = 81\].

Step 6: Calculate the fifth term

To find the fifth term \(a_5\), substitute \(n = 5\) into the formula: \[a_5 = 3^5 = 243\].

Key Concepts

Exponential Growth

Exponential growth is a mathematical concept where a quantity grows at a rate proportional to its current value. In simple terms, the bigger something is, the faster it grows. This is often seen in populations, investments, and of course, numbers in sequences like the one in our exercise.
In this exercise, we observe the sequence generated by the formula \(a_{n} = 3^{n}\). As \(n\) increases, each term becomes the result of multiplying the base number 3 raised to higher powers of \(n\). This leads to a rapid increase, or exponential growth, in the sequence.

• The first term \(a_1 = 3\)
• The second term \(a_2 = 9\)
• The third term \(a_3 = 27\)
• The fourth term \(a_4 = 81\)
• The fifth term \(a_5 = 243\)

These values illustrate how quickly exponential growth can escalate. Each term is significantly larger than the term before it because of the multiplying effect of the powers.

Sequence Formula

A sequence formula, such as \(a_{n} = 3^{n}\), helps us determine each term in a sequence based on its position, \(n\). The clearer and more precise the formula, the easier it is to generate terms without guessing.
In our case, the formula is an exponential one, meaning it includes a base, 3, and an exponent, \(n\). This formula tells us exactly how to calculate each term by taking the base to the power of the term's position. Understanding the role of both the base and the exponent is crucial:

• The base (3) determines the ratio between terms.
• The exponent (n) indicates the term's position within the sequence.

Each substitution of a number for \(n\) gives you a new term in the sequence. The formula makes it easy to keep track of what each term will be, ensuring you never miss a beat in calculating.

Powers of Numbers

The concept of powers, or exponents, is a cornerstone of mathematics that deals with repeated multiplication of a number by itself. In the formula \(a_{n} = 3^{n}\), 3 is the base, and \(n\) is the exponent, dictating how many times the base is multiplied.
When you see \(3^n\), it means:

• \(3^1\) is 3 multiplied by itself once, which is 3.
• \(3^2\) is 3 multiplied by 3, which equals 9.
• \(3^3\) is 3 multiplied by 3 and then by another 3, totaling 27.

This process continues as \(n\) increases. Understanding powers is key to mastering topics like exponential growth and sequence formulas. It shows how a small starting number, like 3, can become extremely large when raised to higher powers, particularly seen in sequences with exponential rules. Powers highlight the potent multiplying effect of the base number.