In an arithmetic progression, the second term is 8, and the 38th term is 224.

In an arithmetic progression, the second term is 8, and the 38th term is 224. Find the difference of this progression and the sum of its 38 first terms.

We find the first term of the progression from the relation a2 = a1 + d:

a1 = a2 – d.

We find the difference d from the formula for the progression term numbered.

an = a1 + d (n – 1);

an = a2 – d + d (n – 1);

an = a2 + d (n – 1 – 1);

an = a2 + d (n – 2);

d = (аn – a2) / (n – 2) = (224 – 8) / (38 – 2) = 216/36 = 6.

Sum of 38 members of the progression:

S = ((a1 + a38) * n) / 2 = ((a2 – d + a38) * n) / 2 = ((8 – 6 + 224) * 38) / 2 = 4294.

Answer. Difference d = 6, sum S = 4294.