What is the sum of the first 28 terms in the following sequence?
1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, ⋯
Answer & explanation
Answer: (b) 84
After the first 1, the sequence runs in blocks 1 | 2, 1 | 3, 2, 1 | 4, 3, 2, 1 | …. The first 28 terms are the lone 1, the full blocks up to 6 (21 terms, sum 56) and the first 6 terms of the block 7, 6, …, 1 (sum 27): 1 + 56 + 27 = 84.
- Group the terms: 1 | 1 | 2, 1 | 3, 2, 1 | 4, 3, 2, 1 | 5, 4, 3, 2, 1 | …; after the first 1, block k is k, k − 1, …, 1.
- Blocks 1 to 6 have 1 + 2 + 3 + 4 + 5 + 6 = 21 terms; with the first 1 that makes 22 terms.
- Their sum = 1 + (1 + 3 + 6 + 10 + 15 + 21) = 1 + 56 = 57.
- Terms 23 to 28 are the first 6 terms of block 7: 7 + 6 + 5 + 4 + 3 + 2 = 27.
- Sum of the first 28 terms = 57 + 27 = 84.
Remember · In block sequences, count terms block by block (1, 2, 3, … terms) and handle the last, partial block separately.
Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·