What is the number of a member of the arithmetic progression equal to 180, if its first term is -20, and the difference is 2.5?

An arithmetic progression is given, in which the first term is a1 = -20, and the difference is d = 2.5.

It is required to determine the number of the member of this progression, equal to 180 (an = 180).

Let’s recall the formula for finding the n-th term of an arithmetic progression:

an = a1 + (n – 1) * d,

where n is the ordinal number of a member of the progression.

From this formula, we express n:

(n – 1) * d = an – a1;

n – 1 = (an – a1) / d;

n = ((an – a1) / d) + 1.

Let’s calculate the number of the member of the progression an = 180:

n = ((180 – (-20)) / 2.5) + 1;

n = ((180 + 20) / 2.5) + 1;

n = 200 / 2.5 + 1;

n = 81.

Answer: n = 81.