To find the explicit formula for a geometric sequence where we know the common ratio and a specific term (in this case, f5 = 81), we start off by recalling the general form of a geometric sequence:
f(n) = a * r^(n-1)
Where:
- f(n) = the nth term of the sequence
- a = the first term of the sequence
- r = the common ratio
- n = the term number
Given that f(5) = 81, we can use this information along with the formula. Substituting n with 5 gives:
f(5) = a * r^(5-1) = a * r^4 = 81
This indicates that the product of the first term and the fourth power of the common ratio equals 81. However, we need the value of the common ratio to find the first term.
Assuming we know the common ratio, we can rearrange the equation to solve for a:
a = 81 / r^4
Substituting this value back into the initial formula gives:
f(n) = (81 / r^4) * r^(n-1)
Hence, the explicit formula for the sequence is:
f(n) = 81 * r^(n-5)
In this formula, r is the common ratio that must be specified along with f(5) = 81 for the sequence. This equation will allow us to calculate any term in the sequence depending on the value of the common ratio.