Work through 2026 AMC 8 Problem 14 on arithmetic sequences with a written solution, hints, and a visual explanation.
AlgebraLevel 3 · AMC-style
Jami picked three equally spaced integers on the number line. The sum of the first and the second numbers is 40, while the sum of the second and third numbers is 60. What is the sum of all three numbers?
A. 70
B. 75
C. 80
D. 85
E. 90
Hints
Let the three numbers be a − d, a, and a + d for some common difference d.
Write the two given sums as equations in a and d, then add the two equations together.
Notice d cancels out when you add the equations — solve directly for a, then find 3a.
Read the step-by-step solution
Answer: B · 75
Set up equations
Let the three numbers be a − d, a, a + d. Then (a − d) + a = 2a − d = 40 and a + (a + d) = 2a + d = 60.
Solve for the sum
Adding the two equations: 4a = 100, so a = 25. The sum = 3a = 75.