Consider the sequence \( \frac{1}{2} ; 4 ; \frac{1}{4} ; 7 ; \frac{1}{8} ; 10 ; \ldots \ldots \) 5.6.1 If the pattern continues in the same way, write down the next TWO terms in the sequence 5.6.2 Calculate the sum of the first 50 terms of the sequence.

The sequence you've provided alternates between two distinct patterns. The first part includes fractions that halve each time: \( \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots \), which would give the next term as \( \frac{1}{16} \). The second part increases by 3 each time: \( 4, 7, 10, \ldots \), leading to adding 3 to 10, giving the next term as \( 13 \). So the next two terms in the sequence are \( \frac{1}{16} \) and \( 13 \). To calculate the sum of the first 50 terms, we can separate the sequence into two parts: the fractions and the integers. For the fractions, they can be expressed as: - First term: \( \frac{1}{2^1}, \frac{1}{2^2}, \frac{1}{2^3}, \ldots, \frac{1}{2^{25}} \) (25 terms) For the integers: - They form an arithmetic sequence: \( 4, 7, 10, \ldots \) with the first term \( a = 4 \) and a common difference \( d = 3 \). - The \( n \)-th term is given by \( a_n = 4 + (n-1) \times 3 = 3n + 1 \). Now, we have: - Sum of the fractions: \( S_{f} = \frac{1/2}{1 - (1/2)} = 1 \) (using the formula for the sum of a geometric series). - Sum of the integers \( S_{i} \) for the first 25 terms: \[ S_{i} = \frac{n}{2} \times (\text{first term} + \text{last term}) = \frac{25}{2} \times (4 + 4 + 3 \cdot 24) = \frac{25}{2} \times (4 + 76) = \frac{25}{2} \times 80 = 25 \times 40 = 1000. \] The total sum of the first 50 terms is: \[ S_{total} = S_{f} + S_{i} = 1 + 1000 = 1001. \] Thus, the sum of the first 50 terms of the sequence is \( 1001 \).