If the first 5 terms of a geometric sequence are , then the
formula for the term in the sequence is

Ask by Coleman Ramirez. in the United States

Jan 25,2025

Question

If the first 5 terms of a geometric sequence are $$ \left\{12,6,3, \frac{3}{2}, \frac{3}{4}ight\} $$, then the formula for the $$ n^{	ext {th }} $$ term in the sequence is

UpStudy AI · Accepted Answer

A geometric sequence is defined by its first term $$ a $$ and a common ratio $$ r $$ such that each subsequent term is obtained by multiplying the previous term by $$ r $$.

Given the first five terms of the sequence:
$$ 12, 6, 3, \frac{3}{2}, \frac{3}{4} $$

1. **Identify the first term ($$ a $$):**
   $$ a = 12 $$

2. **Determine the common ratio ($$ r $$):**
   $$ r = \frac{6}{12} = \frac{1}{2} $$
   To confirm, check subsequent terms:
   $$ \frac{3}{6} = \frac{1}{2}, \quad \frac{\frac{3}{2}}{3} = \frac{1}{2}, \quad \frac{\frac{3}{4}}{\frac{3}{2}} = \frac{1}{2} $$
   So, $$ r = \frac{1}{2} $$ is consistent.

3. **Formulate the $$ n^{\text{th}} $$ term:**
   The general formula for the $$ n^{\text{th}} $$ term of a geometric sequence is:
   $$ a_n = a \cdot r^{n-1} $$
   Substituting the known values:
   $$ a_n = 12 \left( \frac{1}{2} \right)^{n-1} $$

**Final Formula:**
$$
a_n = 12 \cdot \left( \frac{1}{2} \right)^{\,n-1}
$$

Answer: After simplifying, the nth term is 12 multiplied by (½)ⁿ⁻¹. Thus,

aₙ = 12 · (1/2)^(n−1)
The $$ n^{\text{th}} $$ term of the sequence is $$ a_n = 12 \cdot \left( \frac{1}{2} \right)^{n-1} $$.

If the first 5 terms of a geometric sequence are , then the
formula for the term in the sequence is

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If the first 5 terms of a geometric sequence are , then the formula for the term in the sequence is