2.3 The sequence \( 3 ; 5 ; 1 ; 6 ; 9 ; 1 ; 12 ; 13 ; 1 ; \ldots \) is given. Determine the sum of the first 75 terms of the sequence.

To determine the sum of the first 75 terms of the sequence \( 3 ; 5 ; 1 ; 6 ; 9 ; 1 ; 12 ; 13 ; 1 ; \ldots \), we first need to identify the pattern in the sequence. Looking closely at the sequence, we can break it down into segments. The non-repeating parts appear to be: - \(3, 5\) (which we'll call the \( 1^{st} \) and \( 2^{nd} \) terms), - \(1\) (which is the \( 3^{rd} \) term and repeats), - \(6, 9\) (which are the \( 4^{th} \) and \( 5^{th} \) terms), - \(1\) (which is the second occurrence and the \( 6^{th} \) term), - \(12, 13\) (which are the \( 7^{th} \) and \( 8^{th} \) terms), - \(1\) (which is the \( 9^{th} \) term and the third occurrence). Let’s map these terms with their positions: - \( 1^{st} \): \( 3 \) - \( 2^{nd} \): \( 5 \) - \( 3^{rd} \): \( 1 \) - \( 4^{th} \): \( 6 \) - \( 5^{th} \): \( 9 \) - \( 6^{th} \): \( 1 \) - \( 7^{th} \): \( 12 \) - \( 8^{th} \): \( 13 \) - \( 9^{th} \): \( 1 \) Notice that after every two numbers of the pattern (\(3, 5\), then \(6, 9\), then \(12, 13\)), we insert a \(1\). The sequence seems to consist of pairs followed by the number \(1\) repeated. We can also observe the pattern for every set of 3 terms: 1. \(3, 5, 1\) 2. \(6, 9, 1\) 3. \(12, 13, 1\) To clarify: - The random numbers \(3, 5\) \(\to\) \(6, 9\) \(\to\) \(12, 13\) suggest that the series increases by following a direct pattern after incorporating the \(1\). Calculating the total contribution of these groups: - The first group contributes \(3 + 5 + 1 = 9\) - The second group contributes \(6 + 9 + 1 = 16\) - The third group contributes \(12 + 13 + 1 = 26\) Continuing this segment sequentially, we can show every segment of \( (a+b+1) \): If we denote the pairs as: - \( (3, 5) \) → \( A_1 = 3 + 5 + 1 = 9 \) - \( (6, 9) \) → \( A_2 = 6 + 9 + 1 = 16 \) - \( (12, 13) \) → \( A_3 = 12 + 13 + 1 = 26 \). These triplets (1/3 cycles) yields the average pattern. Each triplet sums up consistently. Next step; we will outline how many full triplets exist in the first \(75\) terms: Since a triplet has \(3\) terms, there will be \( \left\lfloor 75 / 3 \right\rfloor = 25\) complete terms. So, the sequence contributions can be figured often: \[ 25 \text{ triplets can be } 1: (3, 5) : 9, (6,9) : 16, (12, 13) : 26 \] If we analyze, we’ve also identified if \(3 \cdot 25 = 75\). Hence, new sum scenarios show (the resulting \(9+16+26\)): Total sum across \( 1^{st}, 2^{nd},... 25^{th}\) The running total will look like: - \(9, 16, 26, ..., = 300\) \( \text{(from }\ 75\text{ summations.)}\) The formula would imply adding multiples. Sum the individual patterns. Thus completing thus, we can assertively conclude \( 25 \cdot 300 = 1200\). So, the sum of the first \(75\) terms of the