An increasing arithmetic progression S (3) = 21, if you subtract 2,3 and 2 from these numbers, respectively

An increasing arithmetic progression S (3) = 21, if you subtract 2,3 and 2 from these numbers, respectively, you get a geometric progression. Find a1, a2, a3.

Let’s use the formula for the sum of an arithmetic progression: Sn = (a1 + an) / 2 * n, where a1, an are the first and nth terms, n is the number of terms. We get the equation:

(a1 + a3) / 2 * 3 = 21;

a1 + a3 = 14.

a2 = a1 + d;

a3 = a1 + 2d.

For a geometric progression, the following formula is valid: bn = b1 * q ^ (n -1). Since b1 = a1 – 2, b2 = a2 – 3, b3 = a3 – 2. We get the equations:

a3 – 2 = (a1 – 2) * q ^ 2.

a2 – 3 = (a1 – 2) * q.

We get a system of 5 equations with 5 unknowns, which has a unique solution.