Given that u12 = 19 and u3 = 1 for an arithmetic sequence with the first term u0 and common difference r, we can find the value of r and u0.
We know that for an arithmetic sequence, the general formula for the nth term is given by:
u_n = u_0 + (n-1)r
Substitute the given values:
u12 = u0 + 11r = 19
u3 = u0 + 2r = 1
Now we have a system of two equations with two variables:
u0 + 11r = 19 (1)
u0 + 2r = 1 (2)
Subtract equation (2) from equation (1):
9r = 18
r = 2
Now substitute the value of r back into equation (2) to find u0:
u0 + 2(2) = 1
u0 + 4 = 1
u0 = -3
Therefore, the common difference (r) is 2 and the first term (u0) is -3 for this arithmetic sequence.
