The difference of the arithmetic progression is 5. The sum of the first eight terms is 1540.

The difference of the arithmetic progression is 5. The sum of the first eight terms is 1540. Find the first term of this progression.

1) The eighth term of this progression (a8) is equal to:

a8 = a1 + (8 – 1) * d = a1 + 7d, where a1 is the first term, d is the difference of the progression.

2) Then the first term can be found like this:

a1 = a8 – 7d.

3) The sum of the first eight members of this progression (S8) is determined by the formula:

S8 = 1/2 * (a1 + a8) * 8;

S8 = 4 * (a1 + a8);

S8 = 4a1 + 4a8.

Since S8 = 1540, then:

4a1 + 4a8 = 1540.

From here

a8 = (1540 – 4a1): 4 = 385 – a1.

4) Find a1:

a1 = a8 – 7d;

a1 = (385 – a1) – 7d;

2a1 = 385 – 7d;

a1 = (385 – 7d): 2.

Since d = 5, then:

a1 = (385 – 7 * 5): 2;

a1 = 175.

Answer: 175.