A geometric sequence starts with the values: an = 7, 21, 63, 189.... What is the exponential equation for this sequence?

Answer 1

To find the exponential equation for the given geometric sequence \( a_n = 7, 21, 63, 189, \ldots \), we start by identifying the first term and the common ratio.

1. Identify the first term \( a_1 \):

   • The first term \( a_1 = 7 \).

2. Identify the common ratio \( r \):

   • To find the common ratio, we divide the second term by the first term: \[ r = \frac{a_2}{a_1} = \frac{21}{7} = 3 \]
   • We can verify this by checking the ratio between subsequent terms: \[ r = \frac{a_3}{a_2} = \frac{63}{21} = 3 \] \[ r = \frac{a_4}{a_3} = \frac{189}{63} = 3 \]
   • The common ratio \( r \) is consistently \( 3 \).

3. Write the general formula for the \( n \)-th term of a geometric sequence: The \( n \)-th term of a geometric sequence can be given by the formula: \[ a_n = a_1 \cdot r^{n-1} \] Substituting the values we found:

   • \( a_1 = 7 \)
   • \( r = 3 \)

   Thus, the equation becomes: \[ a_n = 7 \cdot 3^{n-1} \]

This is the exponential equation for the given geometric sequence.

Answer 2

y = BLANK (BLANK)x

Answer 3

From your prompt, it seems you want the exponential equation in the format \( y = A \cdot B^x \).

Given the sequence we discussed, the equation we derived is:

\[ a_n = 7 \cdot 3^{n-1} \]

To write it in the format \( y = A \cdot B^x \), we can make a substitution for \( n \):

Let \( x = n - 1 \). Then \( n = x + 1 \), and we can rewrite the equation:

\[ a_n = 7 \cdot 3^{(x + 1) - 1} = 7 \cdot 3^x \]

Thus, the equation in the desired format is:

\[ y = 7 \cdot 3^x \]

So we can fill in your blanks as:

\( y = 7 \cdot (3^x) \)

Where \( A = 7 \) and \( B = 3 \).