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Chapter 8: Problem 78

Explain how to find the general term of a geometric sequence.

Short Answer

Expert verified
The general term of a geometric sequence is given by the formula \(a * r^{n-1}\), where \(a\) is the first term, \(r\) is the common ratio and \(n\) is the term number. To find any term, substitute the appropriate values into the formula.

Step by step solution

01

Recognize the Pattern

First, recognize the pattern followed in the geometric sequence. After the first term, each succeeding term is obtained by multiplying the previous term by the same non-zero number. This number is called the common ratio \(r\). The first term is denoted by \(a\).
02

General Formulation

Formulate the general term of the sequence, denoted as \(nth\) term. In a geometric sequence, the \(nth\) term is calculated using the formula \( a * r^{n-1}\). Here, \(a\) represents the first term of the sequence, \(r\) is the common ratio, and \(n\) is the term number.
03

Application

Finally, to find any term in the sequence, substitute the values of \(a\), \(r\), and \(n\) in the formula \(a * r^{n-1}\)

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Most popular questions from this chapter

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ (x+3)^{8} $$

Show that $$ \left(\begin{array}{l}n \\\r\end{array}\right)+\left(\begin{array}{c}n \\\r+1\end{array}\right)=\left(\begin{array}{l}n+1 \\\r+1 \end{array}\right) $$ Hints: $$ \begin{aligned}&(n-r) !=(n-r)(n-r-1) !\\\&(r+1) !=(r+1) r !\end{aligned} $$

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