Question
The table shows the ordered pairs of a geometric sequence.
Which equation represents the value of the 18th term in the geometric sequence?
Responses
A sub 18 is equal to 2 times 18 cubed
A sub 18 is equal to 2 times 3 raised to the 18 minus 1 power
A sub 18 is equal to 2 times 3 plus 18
Which equation represents the value of the 18th term in the geometric sequence?
Responses
A sub 18 is equal to 2 times 18 cubed
A sub 18 is equal to 2 times 3 raised to the 18 minus 1 power
A sub 18 is equal to 2 times 3 plus 18
Answers
GPT-4o mini
To find the equation that represents the value of the 18th term in a geometric sequence, we first need to identify the general formula for the \( n \)-th term of a geometric sequence, which is:
\[ a_n = a_1 \cdot r^{(n-1)} \]
where:
- \( a_n \) is the \( n \)-th term,
- \( a_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.
If we assume from the options given that the first term \( a_1 = 2 \) and the common ratio \( r = 3 \), then the formula for the \( n \)-th term becomes:
\[ a_n = 2 \cdot 3^{(n-1)} \]
For the 18th term (\( n = 18 \)):
\[ a_{18} = 2 \cdot 3^{(18-1)} = 2 \cdot 3^{17} \]
From the options provided, the equation that matches this form is:
**B**: \( A_{18} = 2 \cdot 3^{(18 - 1)} \)
So the correct answer is:
**A sub 18 is equal to 2 times 3 raised to the 18 minus 1 power.**
\[ a_n = a_1 \cdot r^{(n-1)} \]
where:
- \( a_n \) is the \( n \)-th term,
- \( a_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.
If we assume from the options given that the first term \( a_1 = 2 \) and the common ratio \( r = 3 \), then the formula for the \( n \)-th term becomes:
\[ a_n = 2 \cdot 3^{(n-1)} \]
For the 18th term (\( n = 18 \)):
\[ a_{18} = 2 \cdot 3^{(18-1)} = 2 \cdot 3^{17} \]
From the options provided, the equation that matches this form is:
**B**: \( A_{18} = 2 \cdot 3^{(18 - 1)} \)
So the correct answer is:
**A sub 18 is equal to 2 times 3 raised to the 18 minus 1 power.**