(d) Write each of the following series in sigma notation: \( \begin{array}{ll}\text { (1) } 2+4+6+8+10+12 & \text { (2) } 3-6+12-24+\ldots \text { (to } 10 \text { terms) } \\ \text { (3) } \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots+\frac{1}{128} & \text { (4) } 5+5+5+5+\ldots \text { (to } n \text { terms) } \\ \text { (5) } 1+8+27+64+125 & \text { (6) } 2+6+12+20+\ldots \text { (to } n \text { terms) } \\ \text { (7) } 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots \text { (to } n \text { terms) } & \text { (8) } \frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\ldots+\frac{24}{25}\end{array} \)

1) \( \sum_{k=1}^{6} 2k \) - This represents the sum of the first six even numbers, where \( k \) starts from 1 and goes up to 6. 2) \( \sum_{k=0}^{9} 3 \cdot (-2)^k \) - This captures the alternating series of multiplying 3 by powers of -2, creating a sequence that alternates signs for ten terms. 3) \( \sum_{k=1}^{7} \frac{1}{2^k} \) - This notation describes a geometric series where each term is half of the previous one, starting from \( \frac{1}{2} \) up to \( \frac{1}{128} \). 4) \( \sum_{k=1}^{n} 5 \) - A straightforward representation of adding 5, \( n \) times, perfect for calculating the total of several fives! 5) \( \sum_{k=1}^{5} k^3 \) - This captures the sum of the cubes of the first five natural numbers, highlighting the beautify of cubic growth! 6) \( \sum_{k=1}^{n} k(k+1) \) - Each term represents the sum of consecutive integers, flaring up a poetic symmetry as you add terms \( k \) times \( (k+1) \). 7) \( \sum_{k=1}^{n} \frac{1}{k} \) - This notation wonderfully captures the harmonic series, illustrating how sums of fractions blend together like a musical crescendo! 8) \( \sum_{k=1}^{24} \frac{k}{k+1} \) - This series showcases a clever shift of fractions leading up to \( \frac{24}{25} \), perfect for capturing the essence of gradual increase in fractions.