15. Consider the sequence \[ \frac{1}{2} ; 4 ; \frac{1}{4}, 7 ; \frac{1}{8}: 10 ; \] a) If the pattern continues in the same was, write down the next Tws terms in the sequence. b) Calculate the sum of the first 50 terms of the sequeace.

To find the next two terms of the sequence, we first observe the pattern. The sequence alternates between fractions that are decreasing by half and integers that increase by 3. The fractions are given by \( \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots \), which leads us to \( \frac{1}{16} \) next. Meanwhile, the integers start at 4, then go to 7, then to 10, revealing that the next integer is \( 10 + 3 = 13 \). So the next two terms will be \( \frac{1}{16}, 13 \). To calculate the sum of the first 50 terms, we can define the sequence better: for the odd terms (the fractions), they can be expressed as \( \frac{1}{2^n} \) for \( n = 1, 2, 3, \ldots \) and for the even terms (the integers), they can be expressed as \( 3n + 1 \) for \( n = 0, 1, 2, \ldots \). The sum can be broken down into two parts: the sum of the first 25 fractions and the sum of the first 25 integers. The sum of the fractions: \[ S_f = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots + \frac{1}{2^{25}} = 1 - \frac{1}{2^{25}} = 1 - 0.0000000298 \approx 1 \] The sum of the integers: \[ S_i = 4 + 7 + 10 + \ldots + 3n + 1 = 3(0+1+\ldots+24) + 25 = 3 \cdot \frac{24 \cdot 25}{2} + 25 = 3 \cdot 300 + 25 = 925 \] Therefore, the total sum of the first 50 terms is roughly: \[ S_{total} = S_f + S_i \approx 1 + 925 = 926 \] So, the sum of the first 50 terms of the sequence is approximately 926.