Answer:
The first term is 3, and the common difference is 2.
Step-by-step explanation:
The nth term of an arithmetic sequence is:
a = a₀ + d (n − 1)
We are given two equations:
a₄ a₁₀ = 189
a₈ = 2 a₃ + 3
Where:
a₃ = a₀ + d (3 − 1) = a₀ + 2d
a₄ = a₀ + d (4 − 1) = a₀ + 3d
a₈ = a₀ + d (8 − 1) = a₀ + 7d
a₁₀ = a₀ + d (10 − 1) = a₀ + 9d
Substituting:
(a₀ + 3d) (a₀ + 9d) = 189
a₀ + 7d = 2 (a₀ + 2d) + 3
Two equations, two variables. First, simplify:
a₀² + 12 a₀ d + 27d² = 189
0 = a₀ − 3d + 3
Solve for a₀ in the second equation and substitute into the first:
a₀ = 3d − 3
(3d − 3)² + 12 (3d − 3) d + 27d² = 189
9d² − 18d + 9 + 36d² − 36d + 27d² = 189
72d² − 54d − 180 = 0
4d² − 3d − 10 = 0
4d² − 3d − 10 = 0
(d − 2) (4d + 5) = 0
d = -1.25 or 2
a₀ = 3d − 3
a₀ = -6.75 or 3
If a₀ = -6.75 and d = -1.25:
a₄ a₁₀ = -10.5 × -18 = 189
a₈ / a₃ = -15.5 / -9.25 = 1 R 6.25
Extraneous solution
If a₀ = 3 and d = 2:
a₄ a₁₀ = 9 × 21 = 189
a₈ / a₃ = 17 / 7 = 2 R 3
The first term is 3, and the common difference is 2.
