Question

Give the first five terms of the given sequence. $$\left\\{b_{n}\right\\}=\left\\{\left(-\frac{3}{2}\right)^{n}\right\\}$$

Short answer

The first five terms are 1, -\(\frac{3}{2}\), \(\frac{9}{4}\), -\(\frac{27}{8}\), and \(\frac{81}{16}\).

Step-by-step solution

Identify the Sequence Type

The given sequence \( b_{n} \) is defined as \( b_{n} = \left(-\frac{3}{2}\right)^{n} \). This is a geometric sequence where the first term is 1, and the common ratio is \(-\frac{3}{2}\).

Calculate the First Term

Substitute \( n = 0 \) into the formula to find the first term: \( b_{0} = \left(-\frac{3}{2}\right)^{0} = 1 \).

Calculate the Second Term

Substitute \( n = 1 \) into the formula to find the second term: \( b_{1} = \left(-\frac{3}{2}\right)^{1} = -\frac{3}{2} \).

Calculate the Third Term

Substitute \( n = 2 \) into the formula to find the third term: \( b_{2} = \left(-\frac{3}{2}\right)^{2} = \left(\frac{9}{4}\right) \).

Calculate the Fourth Term

Substitute \( n = 3 \) into the formula to find the fourth term: \( b_{3} = \left(-\frac{3}{2}\right)^{3} = -\frac{27}{8} \).

Calculate the Fifth Term

Substitute \( n = 4 \) into the formula to find the fifth term: \( b_{4} = \left(-\frac{3}{2}\right)^{4} = \frac{81}{16} \).

Key concepts

Sequence

A sequence is an ordered collection of numbers that follow a particular pattern. It's like a list of numbers arranged in a specific order. In mathematics, sequences are represented using symbols like \( a_n \) or \( b_n \), where \( n \) is a positive integer telling us the position of a particular term within the sequence. For example, in the sequence where numbers are written as \( \left(-\frac{3}{2}\right)^n \), each term relates to an increasing power of \( \left(-\frac{3}{2}\right) \). This specific organization allows us to predict future terms using a formula, making sequences a key topic in understanding patterns and structures in mathematics.

• Ordered: Sequences are arranged in a particular order based on a rule or formula.
• Pattern: Every sequence follows some identifiable pattern, which matches a specific rule.
• Position: Each element in a sequence is associated with a position number, indicated usually by \( n \).

By understanding sequences, you can unravel the mysteries of their patterns and how each term builds on the one before it.

Common Ratio

The common ratio is the key feature that characterizes geometric sequences. It is the factor by which each term of the geometric sequence is multiplied to get the next term. In a geometric sequence like \( b_{n} = \left(-\frac{3}{2}\right)^n \), the common ratio is \(-\frac{3}{2}\). This means that to move from one term to the next, you multiply the current term by this ratio.
Understanding the common ratio helps you predict the flow of the sequence. Here's why it's essential:

• Predictability: The common ratio allows us to calculate any term in the sequence if we know the previous one.
• Consistency: The sequence remains geometrically coherent because each term relates consistently through multiplication by the common ratio.
• Simplification: Instead of calculating every term from scratch, use the common ratio to move quickly between terms.

By identifying the common ratio, you unlock the prediction power of the entire sequence, making it easier to find any term's value.

Terms

Terms are the individual elements that make up a sequence. Each term has a position, and in mathematical notation, we often refer to terms like \( a_1, a_2, a_3, \) and so on. In the geometric sequence \( \left\{b_{n}\right\} = \left\{\left(-\frac{3}{2}\right)^{n}\right\} \), terms are found using the base \( \left(-\frac{3}{2}\right) \) raised to increasing powers, beginning at zero.

Terms in a sequence have unique properties:

• Distinct: Each term is a distinct number, calculated by its position in the sequence.
• Dependent: Every term depends on the position \( n \) and the sequence's particular rule or formula.
• Incremental: For a geometric sequence, each term is generated by multiplying the previous term by the common ratio.

Understanding the terms of a sequence is crucial because the terms are the sequence's building blocks, helping to provide a clear image of how the sequence progresses.

Geometric Progression

A geometric progression is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The sequence given by \( b_{n} = \left(-\frac{3}{2}\right)^n \) exemplifies a geometric progression since each term is obtained from the previous one using the common ratio \(-\frac{3}{2}\).
Geometric progressions feature prominently in both mathematics and real-life scenarios, due to their logical structure:

• Consistent Growth or Decay: Since each term is a fixed multiple of the one before, geometric progressions can represent steady growth (e.g., compound interest) or decay (e.g., radioactive decay).
• Simple Formula: Utilizing the formula \( a_n = a \cdot r^{n-1} \), where \( a \) is the first term and \( r \) is the common ratio, you can quickly calculate any term.
• Predictability: Once you understand the sequence's first term and common ratio, predicting future terms becomes straightforward and quick.

Grasping the concept of geometric progression empowers you to solve problems involving sequences efficiently and with clarity.