How do you write a formula for the general term (the nth term) of the geometric sequence 4,4/3,4/27,4/81...?

1 Answer
Dec 5, 2015

Answer:

#4/(3^(n-1))#

Explanation:

The formula for the general term should be of the form;

#ar^(n-1)#

Where #a# is a constant, #r# is the rate at which the series is increasing, and #n# is the number of the term. The constant #a# should be the first term, in this case;

#a=4#

We can find #r# by dividing any term by the previous one. Lets use the first two.

#r=(4"/"3)/4 color(white)(color(black)("(2nd term)")/ color(black)("(1st term)")) = 1/3#

Now that we have #a# and #r# we can write our general formula.

#ar^(n-1) = 4(1/3)^(n-1)#

We should double check that we get the correct series. Plugging in the first five #n# values, we get;

#4(1/3)^(1-1), 4(1/3)^(2-1), 4(1/3)^(3-1), 4(1/3)^(4-1), 4(1/3)^(5-1)...#

#=4/3^0, 4/3^1, 4/3^2, 4/3^3, 4/3^4...#

#=4, 4/3, 4/9, 4/27, 4/81...#

Looks like it works!