Question

Find the exponential equation for the sequence: a_n = 7, 21, 63, 189…

y = ?(?)^x

Answer 1

To find the exponential equation for the sequence \( a_n = 7, 21, 63, 189, \ldots \), we can start by considering the relationship between the terms in the sequence.

We notice that:
- \( a_1 = 7 \)
- \( a_2 = 21 \)
- \( a_3 = 63 \)
- \( a_4 = 189 \)

We can express these terms in a general form. First, we can calculate the ratios of consecutive terms:

\[
\frac{a_2}{a_1} = \frac{21}{7} = 3
\]

\[
\frac{a_3}{a_2} = \frac{63}{21} = 3
\]

\[
\frac{a_4}{a_3} = \frac{189}{63} = 3
\]

From this, we see that each term is multiplied by 3 to get the next term, suggesting that this is a geometric (exponential) sequence.

We can express \( a_n \) in the form:

\[
a_n = a_1 \cdot r^{n-1}
\]

where \( a_1 = 7 \) and the common ratio \( r = 3 \). Thus,

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

To express this in the form \( y = a \cdot b^x \), we can set \( x = n - 1 \), which gives us:

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

Therefore, the exponential equation for the sequence is:

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