The fifth term of a geometric sequence is 48 and the ninth term is 768. What is the first term?

1 Answer
Nov 6, 2015

Answer:

The first term is 3.

Explanation:

The general form of a geometric sequence with the first term #a# is #a, ar, ar^2, ar^3, ...# where #r# is the common ratio between terms.

Note that the general form for the #n#th term is #ar^(n-1)#.
Using that with our knowledge of the fifth and ninth terms, we have #ar^4 = 48# and #ar^8 = 768#

We can now eliminate #a# by dividing to obtain
#(ar^8)/(ar^4) = 768/48#
#=> r^4 = 16#

We could solve for #r# by taking the fourth root of 16, but that is not necessary for our goal of finding a.

Now that we have #r^4# we can divide once again to find #a#.
#(ar^4)/r^4 = 48/16#

#=> a = 3#