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

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Answer 1 · 2015-07-17T18:33:00

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

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