A geometric sequence is a sequence in which the ratio of consecutive terms is constant. This constant ratio is called the common ratio. Assuming we have a sequence,
a1, a2, a3, a4, ....an
The first term is a1
Common ratio, r = a2/a1 = a3/a2 = a4/a3
The formula for determining the nth term of a geometric sequence is expressed as
an = a1r^(n - 1)
where
a1 is the first term
r is the common ratio
n is the number of terms
An example of such a sequence would be
3, 9, 27, 243.........
r = 9/3 = 37/9 = 3
To find the explicit formula, we would substitute r = 3 and a1 = 3 into the formula. The explicit formula for this sequence would be
an = 3 * 3^(n - 1)
Thus, for the 21st term, we would substitute n = 21 into the explicit formula. we have
a21 = 3 * 3^(21 - 1)
a21 = 3 * 3^20
a21 = 10460353203
Thus, the 21st term of the sequence is 10460353203