Find the sum of the first six terms of the arithmetic progression if the first term is 7 and the eighth is 42.

1. For a given arithmetic progression A (n), its terms are known, its members A1 = 7 and A8 = 42;

2. According to the formula for determining any member of the progression:

An = A1 + d * (n – 1);

A8 = A1 + d * (8 – 1) = A1 + d * 7 = 42;

d = (42 – A1) / 7 = (42 – 7) / 7 = 5;

3. To calculate the sum of the first six terms of the arithmetic progression, we use the formula:

Sn = ((A1 + An) * n) / 2 = (2 * A1 + d * (n – 1)) * n / 2;

S6 = (2 * 7 + 5 * (6 – 1)) * 6) / 2 = (14 + 25) * 3 = 117.

Answer: The sum of six members of the arithmetic progression is 117.