What is the 5 th term of the
geometric sequence below?

Ask by Blake Bowers. in the United States

Mar 24,2025

Question

What is the 5 th term of the geometric sequence below? $$ f(1)=45, f(n)=\frac{1}{3} f(n-1) $$

UpStudy AI · Accepted Answer

1. The first term of the sequence is given as $$ f(1)=45 $$.

2. The sequence is defined recursively by $$ f(n)=\frac{1}{3}f(n-1) $$. This means that each term is multiplied by $$ \frac{1}{3} $$ to obtain the next term.

3. To find the 5th term, we use the formula for the $$ n $$th term of a geometric sequence:
   $$
   f(n)=f(1)\left(\frac{1}{3}\right)^{n-1}
   $$
   When $$ n=5 $$, we have:
   $$
   f(5)=45\left(\frac{1}{3}\right)^{5-1}=45\left(\frac{1}{3}\right)^4
   $$

4. Calculate $$ \left(\frac{1}{3}\right)^4 $$:
   $$
   \left(\frac{1}{3}\right)^4=\frac{1}{81}
   $$

5. Substitute back into the formula:
   $$
   f(5)=45\left(\frac{1}{81}\right)=\frac{45}{81}
   $$

6. Simplify the fraction $$ \frac{45}{81} $$ by dividing the numerator and the denominator by their greatest common divisor (which is 9):
   $$
   \frac{45 \div 9}{81 \div 9}=\frac{5}{9}
   $$

Thus, the 5th term of the sequence is $$ \frac{5}{9} $$.
The 5th term of the sequence is $$ \frac{5}{9} $$.

What is the 5 th term of the
geometric sequence below?

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