In an arithmetic progression, the sum of the first eight terms is 32, and the sum

In an arithmetic progression, the sum of the first eight terms is 32, and the sum of the first twenty terms is 200. What is the sum of the first 28 terms in this progression?

Given: (an) – arithmetic progression;

S8 = 32; S20 = 200;

Find: S28 -?

The sum of the first n terms of the arithmetic progression is found by the formula: Sn = (a1 + an) * n / 2, so S8 = (a1 + a8) * 8/2 = 32, hence

a1 + a8 = 32/8 * 2 = 8.

S20 = (a1 + a20) * 20/2 = 200, hence a1 + a20 = 200/20 * 2 = 20.

The formula for the nth member of the arithmetic progression: an = a1 + d * (n – 1), where a1 is the first member of the arithmetic progression, d is the difference of the progression, n is the number of its members.

Let’s express a8 and a20:

a8 = a1 + d * (8 – 1) = a1 + 7d,

a20 = a1 + d * (20 – 1) = a1 + 19d.

That. a1 + a8 = a1 + a1 + 7d = 2a1 + 7d, a1 + a20 = a1 + a1 + 19d = 2a1 + 19d.

Let’s compose and solve the system of equations:

2a1 + 7d = 8, (1)

2a1 + 19d = 20 (2)

Let us express from (1) the equations 2a1 = 8 – 7d, substitute this expression into (2) the equation: 8 – 7d + 19d = 20, hence 12d = 12 and, therefore, d = 1.

We substitute the obtained value of d into the equation (1) and find a1:

2a1 + 7 * 1 = 8;

2a1 = 1;

a1 = 0.5.

Find a28 = a1 + 27d = 0.5 + 27 = 27.5.

Now let’s calculate S28 = (a1 + a28) * 28/2 = (0.5 + 27.5) * 28/2 = 392.

Answer: S28 = 392.