Can a Series be the Sum of Two Different Sequences?

📌 CONCEPT: A series can be represented as the sum of two different arithmetic sequences, but this representation may not be unique.

📐 RULE / FORMULA: The sum of two arithmetic sequences can be represented as S = (a1 + a2)/2 * (n1 + n2) where a1, a2 are the first terms and n1, n2 are the number of terms in the two sequences.

💡 WORKED EXAMPLE: Consider the series S = 4 + 7 + 10 + 13 + 16 + ... + 99. We can represent this series as the sum of two arithmetic sequences: S = (2 + 52)/2 * (5 + 20) = 27 * 25. Here, the first sequence has a1 = 2, d = 5, and n = 5, while the second sequence has a1 = 52, d = 5, and n = 20.

⚠️ COMMON MISTAKE: Students often assume that the representation of a series as the sum of two arithmetic sequences is always unique, when in fact it may not be. This can lead to incorrect conclusions about the nature of the series.