What is the sum of the first 13 terms of the arithmetic progression (6, 11, 16, 21, ...)?

468

Direct answer: option A, 468. An arithmetic progression (AP) is a sequence in which the difference between consecutive terms remains constant. Here, the first term is a = 6, the common difference is d = 11 − 6 = 5, and the number of terms is n = 13. Use the sum formula Sₙ = n/2[2a + (n − 1)d]. Therefore, S₁₃ = 13/2[2(6) + (13 − 1)5] = 13/2[12 + 60] = 13/2 × 72 = 13 × 36 = 468. A second method confirms this: the thirteenth term is a₁₃ = a + 12d = 6 + 60 = 66, so the average term is (6 + 66)/2 = 36 and the sum is 13 × 36 = 468. A, 468, is correct because both methods agree. B, 474, is too large and does not result from the formula. C, 486, and D, 492, also fail the same calculation. A useful memory cue is: for n terms, there are n − 1 common-difference steps; do not use n instead of n − 1. Also, the nth-term formula finds one term, whereas the sum formula finds all n terms together.

This is a Class 10 Mathematics question. Chapter: Arithmetic Progressions (AP). Topic: Finding the sum of the first $n$ terms of an AP.