Find the sum of the first fifteen terms of the arithmetic progression if its third term is (-5) and the sixth term is 2.5.

We have an arithmetic progression, for which the values of two terms are known – the third and the sixth:

a3 = -5;

a6 = 2.5.

Let’s find the sum of the first fifteen members.

The formula for the nth term of the arithmetic progression is:

an = a1 + d * (n – 1);

Let’s write this formula for the third and sixth terms:

a3 = a1 + 2 * d;

a6 = a1 + 5 * d;

Subtract the third term from the sixth value:

a6 – a3 = 5 * d – 2 * d = 3 * d;

2.5 – (-5) = 3 * d;

d = 2.5.

Find the first and fifteenth terms:

a1 = a3 – 2 * d = -5 – 2 * 2.5 = -10;

a15 = a1 + 14 * d = -10 + 35 = 25.

S15 = (a1 + a15) * 15/2;

S15 = 7.5 * (25 – 10);

S15 = 112.5.