Problem 4 - Algebra and Trigonometry

Question

write the first four terms of each sequence whose general term is given. $$ a_{n}=\left(\frac{1}{3}\right)^{n} $$

Step-by-Step Solution

Answer

The first four terms of the sequence are \(\frac{1}{3}\), \(\frac{1}{9}\), \(\frac{1}{27}\), and \(\frac{1}{81}\) respectively.

Step 1: Identify the General Term

The general term of the sequence is given by the formula \(a_{n}=(\frac{1}{3})^{n}\). This means that for any term \(a_{n}\), the value is calculated by raising \(1/3\) to the power of 'n', where 'n' is the position of the term in the sequence.

Step 2: Calculate the First term

Substitute \(n=1\) into the formula to find the first term: \(a_{1}=(\frac{1}{3})^{1} = \frac{1}{3}\)

Step 3: Calculate the Second term

Substitute \(n=2\) into the formula to find the second term: \(a_{2}=(\frac{1}{3})^{2} = \frac{1}{9}\)

Step 4: Calculate the Third term

Substitute \(n=3\) into the formula to find the third term: \(a_{3}=(\frac{1}{3})^{3} = \frac{1}{27}\)

Step 5: Calculate the Fourth term

Substitute \(n=4\) into the formula to find the fourth term: \(a_{4}=(\frac{1}{3})^{4} = \frac{1}{81}\)

Key Concepts

General Term

The concept of a general term in a geometric sequence is essential for understanding how each term in the sequence is generated. A general term, often represented as \(a_n\), is a formula that describes each term in the sequence based on its position. In the exercise, the general term is given as \(a_n = \left(\frac{1}{3}\right)^n\). This means each term is found by raising the fraction \(\frac{1}{3}\) to the power of the term's position, indicated by \(n\). Understanding the general term helps to systematically calculate any term in the sequence without listing all previous terms.

Sequence Terms

Sequence terms refer to the individual components within a sequence. In a geometric sequence, each term is generated by applying the general term formula to successive integers. Let's explore the first four terms as described by the formula \(a_n = \left(\frac{1}{3}\right)^n\):

• For \(n=1\), the first term, \(a_1 = \left(\frac{1}{3}\right)^1 = \frac{1}{3}\)
• For \(n=2\), the second term, \(a_2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9}\)
• For \(n=3\), the third term, \(a_3 = \left(\frac{1}{3}\right)^3 = \frac{1}{27}\)
• For \(n=4\), the fourth term, \(a_4 = \left(\frac{1}{3}\right)^4 = \frac{1}{81}\)

Knowing how to find these terms is crucial for mastering geometric sequences.

Exponentiation

Exponentiation is a key mathematical operation used to represent repeated multiplication. In the general term \(a_n = \left(\frac{1}{3}\right)^n\), the exponent \(n\) indicates how many times \(\frac{1}{3}\) is multiplied by itself. For instance, \(\left(\frac{1}{3}\right)^2 = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}\). Exponentiation is essential in geometric sequences because it creates the consistent ratio between terms. The operation can have significant effects depending on the base value and the size of the exponent, making a solid understanding of it important in mathematics.

Mathematical Calculation

Mathematical calculation in geometric sequences involves systematically applying operations to find specific terms. The steps are straightforward yet require careful attention to detail:

• Identify the general term: This is the formula used for calculations.
• Substitute the position: Replace \(n\) with the term number required.
• Calculate: Use basic arithmetic operations to find the result.

For instance, calculating \(a_2\) means substituting \(n = 2\) in \(a_n = \left(\frac{1}{3}\right)^n\), which gives \(\left(\frac{1}{3}\right)^2 = \frac{1}{9}\). Mastering these steps allows students to efficiently work through sequences and understand mathematics more deeply.