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QUESTION

# 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

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"