In geometric progression (A n) a 10 = 27, a12 = 108. find a11.

In geometric progression (A n) it is known:

a10 = 27;
a12 = 108.
Let’s find a11.

a (n + 1) = an * q;

a11 = a10 * q;

a12 = a11 * q;

Let’s compose a system of equations. For this, we substitute the known values into the expressions and calculate a11 of the geometric progression.

{a11 = 27 * q;

108 = a11 * q;

{27 * q = a11;

a11 * q = 108;

{q = a11 / 27;

q = 108 / a11;

From here we get:

q = q;

a11 / 27 = 108 / a11;

108 * 27 = a11 * a11;

(a11) ^ 2 = 108 * 27;

(a11) ^ 2 = 324 * 9;

a11 = √ (324 * 9);

a11 = √324 * √9;

a11 = √18 ^ 2 * √3 ^ 2;

a11 = 18 * 3;

a11 = 54.