The pattern in the given sequence is that the numerators are increasing by powers of 4, while the denominators are increasing by powers of 81.
To find the nth term in this sequence, we can use the formula an = a1 * r^(n-1), where an is the nth term, a1 is the first term, and r is the common ratio. In this sequence, a1 = 1/3 and r = 4/81.
The common ratio in this sequence is 4/81.
To simplify the given sequence, we can first rewrite the fractions as powers of 3 and 81. This gives us 1/3 = 3^-1 and 4/81 = 3^-4. Then, we can combine the like terms to get 3^-1 + 3^-4 + 3^-2 + 3^-8. Finally, using the rule a^-b = 1/a^b, we can simplify the sequence to 3^-1 + 3^-4 + 3^-2 + 3^-8 = 1/3 + 1/81 + 1/9 + 1/59049.
The general formula for the nth term in a sequence where the numerators increase by powers of x and the denominators increase by powers of y is an = a1 * (x/y)^(n-1). In the given sequence, x = 4 and y = 81, so the formula becomes an = a1 * (4/81)^(n-1).