The sum of the fifth and ninth terms of the arithmetic progression is 12. Find the sum of the thirteen

The sum of the fifth and ninth terms of the arithmetic progression is 12. Find the sum of the thirteen first terms of the arithmetic progression.

1. For a given arithmetic progression A (n), the sum of its members:

A5 + A9 = 12;

2. It is required to find the sum of the first thirteen members of the progression:

S13 = ((2 * A1 + d * (13 – 1)) / 2) * 13 =

((2 * A1 + 12 * d) / 2) * 13;

3. Determine the sum of two terms:

A5 + A9 = (A1 + 4 * d) + (A1 + 8 * d) = 2 * A1 + 12 * d = 12;

(what we need, now it becomes clear why these members are taken)

S13 = ((2 * A1 + 12 * d) / 2) * 13 = 12/2 * 13 = 6 * 13 = 78.

Answer: the sum of the first thirteen members is arithmetic. the progression is 78.