Two sums name the progression

A list's first 5 terms total 30. Its first 13 terms total 182. Which list is it?

The list is 2, 4, 6, 8, …, the even numbers. Here is the working.

Write each total with the other face, Sₙ = n/2 (2a + (n − 1)d). Tidy it to an equation in a and d.

5/2 (2a + 4d) = 30 gives 2a + 4d = 12, so a + 2d = 6.

13/2 (2a + 12d) = 182 gives 2a + 12d = 28, so a + 6d = 14.

Rule: two totals, for different counts of terms, give two linear equations. Subtract one from the other to remove a letter. Scale first if no letter has the same number in both. This method is elimination. Linear means a and d have no powers.

Subtract the first equation from the second: (a + 6d) − (a + 2d) = 14 − 6. So 4d = 8 and d = 2. Then a + 2 × 2 = 6 gives a = 2. The list is 2, 4, 6, 8, … — the even numbers. Check: 2 + 4 + 6 + 8 + 10 = 30.

Slide a and d. Match both totals.