Finding the n th Term of a Sequence In Exercises 37-48,...

Finding the $n$ th Term of a Sequence In Exercises
$37-48,$ write an expression for the apparent $n$ th term
$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$
$$
\frac{1}{2},-\frac{1}{4}, \frac{1}{8},-\frac{1}{16}, \ldots
$$

Key Concepts

Sequence

A sequence is an ordered collection of numbers that follow a specific rule or pattern. Sequences are foundational in mathematics for studying patterns, series, and functions, and understanding how each term is generated from its position is crucial.

Geometric Sequence

A geometric sequence is a type of sequence in which each term is obtained by multiplying the previous term by a constant factor known as the common ratio. This type of sequence demonstrates exponential behavior where the growth or decay of values depends on the magnitude of the common ratio.

Common Ratio

The common ratio is the constant factor between consecutive terms in a geometric sequence. It is determined by dividing any term by the term before it and is essential for defining the series' progression and for formulating the nth term.

nth Term Formula

The nth term formula provides a method to directly calculate any term in a sequence without listing all the preceding terms. For a geometric sequence, the nth term is given by a(n) = a(1) multiplied by the common ratio raised to the power of (n - 1), which encapsulates both the multiplicative pattern and the starting term.

Alternating Signs

Alternating signs occur when consecutive terms in a sequence switch between positive and negative. This pattern is often incorporated by using a factor such as (-1) raised to a power dependent on the term index, ensuring that the sign alternates in a regular manner throughout the sequence.

Transcript

Please give Ace some feedback