The first term of a geometric sequence is 5 and the common ratio is 0.5. What is the fourth term?
Understand the Problem
The question is asking to find the fourth term of a geometric sequence given the first term and the common ratio.
Answer
The fourth term is $0.625$.
Answer for screen readers
The fourth term of the geometric sequence is $0.625$.
Steps to Solve
- Identify the terms of the geometric sequence
The first term ($a_1$) is 5 and the common ratio ($r$) is 0.5.
- Determine the formula for the nth term
The formula for the nth term of a geometric sequence is given by: $$ a_n = a_1 \cdot r^{(n-1)} $$
- Calculate the fourth term
Substituting $n = 4$, $a_1 = 5$, and $r = 0.5$ into the formula: $$ a_4 = 5 \cdot (0.5)^{(4-1)} = 5 \cdot (0.5)^3 $$
- Compute the value of $(0.5)^3$
Calculate: $$ (0.5)^3 = 0.125 $$
- Final Calculation for the fourth term
Now substitute back into the equation for $a_4$: $$ a_4 = 5 \cdot 0.125 = 0.625 $$
The fourth term of the geometric sequence is $0.625$.
More Information
In geometric sequences, each term is found by multiplying the previous term by a fixed ratio. Since the ratio is less than one, the terms will decrease.
Tips
- Forgetting to adjust the exponent based on the term number. Make sure to use $(n-1)$ for the formula.
- Miscalculating powers of the common ratio. Double-check calculations when computing powers of fractions.
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