Write an equation to describe the sequence below. Use n to represent the position of a term in the sequence, where n = 1 for the first term. 3.51, 17.55, 87.75, ... Write your answ... Write an equation to describe the sequence below. Use n to represent the position of a term in the sequence, where n = 1 for the first term. 3.51, 17.55, 87.75, ... Write your answer using decimals and integers. a_n = ____ (____)^n - 1
Understand the Problem
The question is asking us to derive a formula that describes a given geometric sequence. The sequence starts with 3.51 and each term seems to be multiplied by a constant factor. We need to identify that factor and express the nth term of the sequence in a specific format.
Answer
$$ a_n = 3.51 \cdot 5^{n-1} $$
Answer for screen readers
The equation that describes the sequence is: $$ a_n = 3.51 \cdot 5^{n-1} $$
Steps to Solve
- Identify the first term
The first term of the sequence is given as ( a_1 = 3.51 ).
- Determine the common ratio
To find the common ratio ( r ), divide the second term by the first term: $$ r = \frac{17.55}{3.51} $$
Calculating this: $$ r \approx 5 $$
- Confirm the common ratio with the next terms
Verify that the ratio holds between subsequent terms: $$ r = \frac{87.75}{17.55} $$ Calculating gives: $$ r \approx 5 $$
- Express the nth term formula
The general formula for the ( n )-th term of a geometric sequence is given by: $$ a_n = a_1 \cdot r^{n-1} $$
Substituting the values we found: $$ a_n = 3.51 \cdot 5^{n-1} $$
The equation that describes the sequence is: $$ a_n = 3.51 \cdot 5^{n-1} $$
More Information
In this geometric sequence, each term is obtained by multiplying the previous term by the constant ratio of 5. The first term is ( 3.51 ), and this formula allows you to find any term in the sequence as you increase ( n ).
Tips
- Confusing a geometric sequence with an arithmetic sequence. Remember, in a geometric sequence, terms are multiplied by a constant factor rather than added.
- Miscalculating the common ratio. Always double-check the divisions to ensure accuracy.
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