Problem 6 - 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: -1/3, 1/9, -1/27, 1/81

Step 1: Substitute n=1

Substitute 'n = 1' in the given general term formula to get the first term of the sequence: \(a_{1}=\left(-\frac{1}{3}\right)^{1}=-\frac{1}{3}\)

Step 2: Substitute n=2

For the second term, replace 'n' with 2 : \(a_{2}=\left(-\frac{1}{3}\right)^{2}=\frac{1}{9}\)

Step 3: Substitute n=3

For the third term, replace 'n' with 3 : \(a_{3}=\left(-\frac{1}{3}\right)^{3}=-\frac{1}{27}\)

Step 4: Substitute n=4

For the fourth term, replace 'n' with 4 : \(a_{4}=\left(-\frac{1}{3}\right)^{4}=\frac{1}{81}\)

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the sequence 2, 6, 18, 54, the common ratio is 3, because each term is three times the term before it.

In the given exercise, the general term of the sequence is expressed as \( a_{n} = \big(-\frac{1}{3}\big)^{n} \). This signifies that the sequence is geometric, as each term can be found by raising the common ratio \( -\frac{1}{3} \) to the power of the term's position in the sequence, \( n \). The negative sign in the common ratio results in alternating signs between terms, which is a characteristic of some geometric sequences.

Sequence Terms Calculation

To calculate the terms of a sequence, especially for geometric sequences, we apply the position of the term, denoted generally as \( n \), to the sequence's general formula. For the sequence given by the general term \( a_{n} = \big(-\frac{1}{3}\big)^{n} \), we compute each term using successive values of \( n \).

Starting with \( n = 1 \) and moving forward, we get consecutive terms by substituting this value of \( n \) into the general term formula. This step-by-step approach ensures accuracy and helps us understand how the sequence develops, one term at a time. It is a systematic method for uncovering the structure of the sequence based on its defining formula.

Exponentiation in Sequences

Exponentiation plays a pivotal role in geometric sequences. Each term in such a sequence involves raising the common ratio to an exponent corresponding to the term's position. In our example exercise, exponentiation is applied to the common ratio \( -\frac{1}{3} \) with exponents representing each term's ordinal number.

Understanding the effects of exponentiation is crucial. When the base is negative, as in \( -\frac{1}{3} \), raising it to an even exponent results in a positive term, whereas odd exponents lead to negative terms. This pattern is observable in the alternating signs of the terms from the exercise, showing a fundamental aspect of exponentiation in such sequences. The power to which the common ratio is raised not only determines the magnitude of each term but also, in cases involving negative bases, the sign of those terms.