Question

Find the indicated term for the following geometric sequence 100,80,64,…a12
12th term=

Answer 1

The common ratio between terms is 4/5, since

80/100 = 64/80 = 45

Since the sequence starts with
`a_1=100`, we have

`a_2 = \frac45 a_1`

`a_2 = \frac45 a_1 = \left(\frac45\right)^2 a_1`

`a_3 = \frac45 a_2 = \left(\frac45\right)^3 a_1`

and so on, up to the n-th term,

`a_n = \left(\frac45\right)^n a_1`

Then the 12th term of the sequence is

`a_(12) = \left(\frac45\right)^(12) * 100 = (2^(24))/(5^(12)) * 2^2*5^2 = \boxed{(2^(26))/(5^(10))} = (67108864)/(9765625)`