How to find missing terms in a geometric sequence?
Understand the Problem
The question is asking how to identify terms that are not known in a geometric sequence, which is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. To solve this, we will need to use the properties of geometric sequences and the formula for the nth term.
Answer
$54$
Answer for screen readers
The value of the unknown term is $54$.
Steps to Solve
- Identify the first term and common ratio
Start by determining the first term of the geometric sequence, denoted as $a_1$, and the common ratio, denoted as $r$. If these values are provided in the problem, note them for use in the subsequent steps.
- Use the formula for the nth term
The nth term of a geometric sequence can be calculated using the formula: $$ a_n = a_1 \cdot r^{(n-1)} $$ Here, $n$ represents the position of the term you want to calculate.
- Plug in known values
If the problem provides specific values for $a_1$, $r$, and $n$, substitute them into the equation from Step 2. For example, if $a_1 = 2$, $r = 3$, and you want to find the 4th term ($n = 4$), you would have: $$ a_4 = 2 \cdot 3^{(4-1)} $$
- Calculate the unknown term
Perform the calculations to find the value of $a_n$. For our example of finding $a_4$: $$ a_4 = 2 \cdot 3^{3} = 2 \cdot 27 = 54 $$
- Write down the final result
State the value of the unknown term clearly, providing the answer to the problem.
The value of the unknown term is $54$.
More Information
Geometric sequences are commonly used in various fields, including finance for calculating compound interest and in computer science for algorithm analysis.
Tips
- Confusing common ratio with common difference: In a geometric sequence, the ratio is constant, not the difference. Always ensure you are dealing with multiplication rather than addition.
- Incorrect exponent: When substituting $n$ into the formula, remember to subtract 1 from $n$ to get the power of $r$.
