Find the sum of the first five terms of the geometric progression, in which y1 = 32 y5 = 2.

Find the denominator q of this progression.

By the condition of the problem, the first term b1 of this geometric progression is 32, and the fifth term b5 of this geometric progression is 2.

Using the formula for the nth term of the geometric progression bn = b1 * q ^ (n – 1) for n = 5, we get the following equation:

32 * q ^ (5 – 1) = 2.

We solve the resulting equation:

q ^ 4 = 2/32;

q ^ 4 = 1/16;

q ^ 4 = (1/2) ^ 4;

q1 = 1/2;

q2 = -1/2.

Let us find the second, third and fourth terms of the progression for q = 1/2:

b2 = b1 * q = 32 * (1/2) = 16;

b3 = b2 * q = 16 * (1/2) = 8;

b4 = b3 * q = 8 * (1/2) = 4.

Find the sum of the first five terms of the geometric progression at q = 1/2:

32 + 16 + 8 + 4 + 2 = 62.

Find the second, third and fourth terms of the progression at q = -1/2:

b2 = b1 * q = 32 * (-1/2) = -16;

b3 = b2 * q = -16 * (-1/2) = 8;

b4 = b3 * q = 8 * (-1/2) = -4.

Find the sum of the first five terms of the geometric progression at q = -1/2:

32 – 16 + 8 – 4 + 2 = 22.

Answer: this amount can have 2 values: 62 and 22.