How do you find the explicit formula and calculate term 20 for 3, 9 , 27, 81, 243?

The explicit formula for the progression is ##color(red)(t_n =3^n)## and ##color(red)(t_20 = "3 486 784 401")##.

This looks like a geometric sequence, so we first find the common ratio ##r## by dividing a term by its preceding term.

Your progression is ##3, 9 , 27, 81, 243##.

##t_2/t_1 = 9/3= 3##

##t_3/t_2 = 27/9= 3##

##t_4/t_3 = 81/27= 3##

##t_5/t_4 = 243/81 = 3##

So ##r = 3##.

The ##n^"th"## term in a geometric progression is given by:

##t_n = ar^(n-1)## where ##a## is the first term and ##r## is the common difference

So, for your progression.

##t_n = ar^(n-1) =3(3)^(n-1) = 3^1 × 3^(n-1) = 3^(n-1+1)##

##t_n =3^n##

If ##n = 20##, then

##t_20 = 3^20 = "3 486 784 401"##