To find the explicit formula for a geometric sequence, we need to understand the components involved. A geometric sequence can be represented using the formula:
an = a1 * r(n-1)
Where:
- an is the nth term of the sequence.
- a1 is the first term of the sequence.
- r is the common ratio.
- n is the term number.
In this case, we know that the 5th term (a5) of the sequence equals 81. Using this information, we can express a5 in terms of the first term and the common ratio:
a5 = a1 * r(5-1) = a1 * r4
Thus, we can simplify this to:
81 = a1 * r4
To determine the explicit formula of the sequence, we require both the first term (a1) and the common ratio (r). If the common ratio is given, we can isolate a1 from the equation:
a1 = 81 / r4
This allows us to substitute back into our original explicit formula:
an = (81 / r4) * r(n-1)
Which simplifies to:
an = 81 * r(n-5)
In summary, provided you know the common ratio (r), you can substitute it into the formula above to obtain the explicit formula for the sequence. For instance, if the common ratio is 3, then:
an = 81 * 3(n-5)
This formula can generate the terms of the sequence based on the value of n you choose.