How many terms of the series 6 + 12 + 18 + 24 + … sum to 1800?

Given data

• Arithmetic progression with a = 6 and d = 6; sum S = 1800.

Concept / Approach

• Use AP sum: S_n = n/2 [2a + (n − 1)d].

Step-by-step

S_n = n/2 [12 + 6(n − 1)] = n/2 [6n + 6] = 3n(n + 1)Set 3n(n + 1) = 1800 ⇒ n(n + 1) = 600n^2 + n − 600 = 0 ⇒ Discriminant = 1 + 2400 = 2401 = 49^2n = (−1 + 49)/2 = 24

Verification

S = 3 × 24 × 25 = 1800 ✔

Common pitfalls

• Forgetting to divide by 2 in the sum formula.

Final Answer

Number of terms = 24.