Originally Posted by

**topsquark** The terms of an arithmetic sequence are:

a_n = a_1 + (n - 1)*d

where a_1 is the first term of the series and d is the separation between the terms.

So

a_6 + a_9 = 20 ==> 2*a_1 + 13d = 20

a_6 * a_9 = 64 ==> (a_1)^2 + 13a_1 * d + 40d^2 = 64

From the first equation we get that

d = (2/13)(10 - a_1)

Inserting this into the second equation:

(a_1)^2 + 13a_1*(2/13)(10 - a_1) + 40(2/13)^2 *

(10 - a_1)^2 = 64

Eventually this reduces to:

9(a_1)^2 - 180*a_1 - 5184 = 0

Thus a_1 = 36 or a_1 = -16 by your favorite method of solving quadratics.

Since a_1 is required to be negative, thus a_1 = -16.

So

d = (2/13)(10 - (-16)) = 4.

Thus

a_{10} = -16 + 9*4 = 20

-Dan