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What is the fifth term of the following sequence?

$$a_n = \left(-\dfrac{1}{2}\right)^{n - 1} $$

Question:

What is the fifth term of the following sequence?

$$a_n = \left(-\dfrac{1}{2}\right)^{n - 1} $$

Sequence:

A sequence is comprised of a finite or infinite number of terms (or items) arranged in a certain order. For example, the {eq}n{/eq}th term of the sequence {eq}3, 5, 7, 9, \dots {/eq} can be expressed as:

$${a_n} = 2n + 1 $$

where {eq}n {/eq} indicates the term number.

If we know the {eq}n{/eq}th term of a sequence, we can easily determine the value of a specific term by directly substituting the term number.

Answer and Explanation:

The {eq}n{/eq}th term of the given sequence is

$${a_n} = {\left( { - \dfrac{1}{2}} \right)^{n - 1}} $$

Substitute {eq}n=5 {/eq} in the given expression to evaluate the fifth term of the sequence.

$$\begin{align*} {a_5} &= {\left( { - \dfrac{1}{2}} \right)^{5 - 1}}\\[0.3cm] &= {\left( { - \dfrac{1}{2}} \right)^4}\\[0.3cm] &= \dfrac{1}{{{{\left( 2 \right)}^4}}}&\left[ \ {{\rm{Use}}\;{\rm{the \ formula}}\;{{\left( { - 1} \right)}^n} = 1\;{\rm{if}}\;n\;{\rm{is}}\;{\rm{even}}} \ \right]\\[0.3cm] &= \dfrac{1}{{16}} \end{align*}\\ $$

Thus, the fifth term of the sequence is {eq}\bf{\dfrac{1}{{16}}} {/eq}.


Learn more about this topic:

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What is a Mathematical Sequence?

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Chapter 12 / Lesson 1
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Learn about the definition of sequence in math. Understand what finite and infinite mathematical sequences are and how they are represented. See examples of famous mathematical sequences that are commonly used.


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