Question

Write an expression for the \(n\) th term of the sequence. (There is more than one correct answer.) \(2,-1, \frac{1}{2},-\frac{1}{4}, \frac{1}{8}, \ldots\)

Short answer

So, the expression for the \(n\)th term of the given sequence is \(T_n = (-1)^n \cdot 2^{3-n}\)

Step-by-step solution

Observing the Pattern

Observing the pattern in the sequence, it can be noticed that it starts with 2 and each subsequent term is half of the previous term and alternates in sign. So, the nth term could be expressed in the form \((-1)^n\) times something, as this would take care of the alternating sign.

Expressing in terms of \(n\)

As per the pattern observed, the \(n\)th term can be expressed as \((-1)^n\) times 2 divided by 2 to the power (\(n\)-1), which takes care of the halving with each step. This gives the nth term expression as \(T_n = (-1)^n \cdot \frac{2}{2^{n-1}}\)

Simplifying the expression

For final simplification, we cancel the 2 in numerator and denominator by subtracting the powers of 2. The expression for \(n\)th term simplifies to \(T_n = (-1)^n \cdot 2^{2-(n-1)} = (-1)^n \cdot 2^{3-n} \)

Key concepts

Mathematical Sequences

When we talk about mathematical sequences, we refer to an ordered list of numbers that follow a particular rule or pattern. Each number in the sequence is called a term. To find the position of any term within a sequence, mathematicians use a formula known as the 'nth term'. This is incredibly useful for understanding how sequences evolve and for predicting future terms without having to manually compute each one.

For example, in our exercise, the sequence starts with 2 and each term after the first is generated by taking half of the prior term and changing its sign. Such sequences, which are defined by a clear rule, can often be described by algebraic expressions. Once we've identified the 'engine' of the sequence—what transforms one term to the next—we can encapsulate this in an nth term expression that would allow us to calculate any term just by knowing its position, denoted by the variable 'n'.

Patterns in Sequences

Patterns in sequences are the underlying rules that dictate the progression from one term to the next. Identifying these patterns is crucial as it enables us to predict future terms and establish their algebraic expressions. Sometimes, sequences have simple patterns, such as arithmetic sequences where the difference between consecutive terms is constant, or geometric sequences where each term is a constant multiple of the previous one.

In this exercise, the sequence exhibits a pattern where each term is half the previous term and alternates in sign. This type of pattern is not as straightforward as arithmetic or geometric patterns but can still be described mathematically. Patterns may involve operations like multiplication or division, powers, or even alternating signs, as in this case. Understanding the nature of these patterns is essential for grasping the behavior of sequences and for formulating their nth term expressions.

Expression Simplification

The process of expression simplification is about making the algebraic or numeric expression as compact or as simple as possible, usually by combining like terms and reducing the number of operations. When dealing with sequences, simplifying the nth term is crucial to make it easier to compute any term in the sequence.

In our provided solution, the nth term expression for the sequence starts with a more complex form \(T_n = (-1)^n \cdot \frac{2}{2^{n-1}}\). After realizing that we can simplify the powers of 2 by subtracting exponents, we get the much cleaner form \(T_n = (-1)^n \cdot 2^{3-n}\). This final expression has fewer components, making it more straightforward to use for calculating terms of the sequence. Simplifying expressions is not just about making the math look neater; it often reveals more about the characteristics of the sequence and reduces computational errors.