17.2 The first term of a geometric sequence is 15 . If the second term is 10 , calculate: 17.2.1 17.2.2 \( T_{10} \) Given: \( \quad S_{9} ;-\frac{1}{2} ; 0 ; \frac{1}{2} ; 0 ; \frac{3}{2} ; 0 ; \frac{5}{2} ; 0 ; \frac{7}{2} ; 0 ; \ldots \) Assume that this number pattern continues consistently. \( \begin{array}{ll}\text { 17.3.1 } & \text { Write down the value of the } 191^{s t} \text { term of this sequence. } \\ \text { 17.3.2 } & \text { Determine the sum of the first } 500 \text { terms of this sequence. }\end{array} \)

Let's solve the problems step by step. ### Problem 17.2: Geometric Sequence **Given:** - First term \( a = 15 \) - Second term \( T_2 = 10 \) **Step 1: Find the common ratio \( r \)** The second term of a geometric sequence can be expressed as: \[ T_2 = a \cdot r \] Substituting the known values: \[ 10 = 15 \cdot r \] To find \( r \), we can rearrange the equation: \[ r = \frac{10}{15} = \frac{2}{3} \] **Step 2: Calculate \( T_{10} \)** The \( n \)-th term of a geometric sequence is given by: \[ T_n = a \cdot r^{n-1} \] For \( n = 10 \): \[ T_{10} = 15 \cdot \left(\frac{2}{3}\right)^{10-1} = 15 \cdot \left(\frac{2}{3}\right)^9 \] Now, let's calculate \( \left(\frac{2}{3}\right)^9 \): \[ \left(\frac{2}{3}\right)^9 = \frac{2^9}{3^9} = \frac{512}{19683} \] Thus, \[ T_{10} = 15 \cdot \frac{512}{19683} = \frac{7680}{19683} \] ### Problem 17.3: Number Pattern **Given:** The sequence is \( S = -\frac{1}{2}, 0, \frac{1}{2}, 0, \frac{3}{2}, 0, \frac{5}{2}, 0, \frac{7}{2}, 0, \ldots \) **Step 1: Identify the pattern** The sequence alternates between a term and zero. The non-zero terms are: \[ -\frac{1}{2}, \frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \frac{7}{2}, \ldots \] The non-zero terms can be expressed as: \[ T_n = \frac{(n-1)}{2} \quad \text{for odd } n \] For even \( n \), the term is \( 0 \). **Step 2: Find the value of the \( 191^{st} \) term** Since \( 191 \) is odd, we use the formula for odd \( n \): \[ T_{191} = \frac{(191-1)}{2} = \frac{190}{2} = 95 \] **Step 3: Determine the sum of the first 500 terms** The first 500 terms consist of 250 non-zero terms and 250 zeros. The non-zero terms are: \[ -\frac{1}{2}, \frac{1}{2}, \frac{3}{2}, \ldots, \frac{249}{2} \] The sum of the first 250 non-zero terms can be calculated as: \[ S = -\frac{1}{2} + \frac{1}{2} + \frac{3}{2} + \ldots + \frac{249}{2} \] This is an arithmetic series where: - First term \( a = -\frac{1}{2} \) - Last term \( l = \frac{249}{2} \) - Number of terms \( n = 250 \) The sum of an arithmetic series is given by: \[ S_n = \frac{n}{2} \cdot (a + l) \] Substituting the values: \[ S_{250} = \frac{250}{2} \cdot \left(-\frac{1}{2} + \frac{249}{2}\right) = 125 \cdot \left(\frac{248}{2}\right) = 125 \cdot 124 = 15500 \] ### Final Answers: - **17.2.1** \( T_{10} = \frac{7680}{19683} \) - **17.3.1** The value of the \( 191^{st} \) term is \( 95 \). - **17.3.2** The sum of the first 500 terms is \( 15500 \).