Find the difference and the first term of the arithmetic progression (аn), a5 = 86 and a17 = 104

Given: (an) – arithmetic progression;

a5 = 86; a17 = 104;

Find: a1 -?, D -?

The formula for the n-th member of the arithmetic progression:

an = a1 + d * (n – 1), where a1 is the first member of the arithmetic progression, d is the difference of the progression, n is the number of its members.

Let us express, according to this formula, the fifth and seventeenth terms of the progression:

a5 = a1 + 4d = 86; a17 = a1 + 16d = 104.

Let’s compose and solve the system of equations:

a1 + 4d = 86, (1)

a1 + 16d = 104 (2)

Let us express from (1) the equation a1, we get: a1 = 86 – 4d.

Substitute the resulting expression into (2) the equation:

86 – 4d + 16d = 104;

12d = 18;

d = 1.5 – we substitute this result into the expression for finding a1 = 86 – 4d = 86 – 4 * 1.5 = 80.

Answer: a1 = 80, d = 1.5.