Find the first negative term of the arithmetic progression 12.5; 11.2; …

Knowing the first and second terms of this arithmetic progression, we find its difference d:

d = 11.2 – 12.5 = -1.3.

Applying the formula for the nth term of the arithmetic progression, we write the formula for the nth term of this sequence:

an = a1 + d * (n – 1) = 12.5 + (-1.3) * (n – 1) = 12.5 – 1.3n + 1.3 = 13.8 – 1.3n.

Find the first first negative term of this sequence by calculating the smallest positive integer solution to the inequality:

13.8 – 1.3n <0;

1.3n> 13.8;

n> 13.8 / 1.3;

n> 138/13;

n = 11.

We find a11:

a11 = 13.8 – 1.3 * 11 = 13.8 – 14.3 = -0.5.

Answer: -0.5.