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"#