2. The first three terms of an inginite geometric sequence are 16,8 and 4 respectively. a) Determine the \( n^{\text {th }} \) term of the sequence. b) Determine all possible values of \( n \), for which the sum of the first \( n \) terms of this sequence is greater than 31 . c) Calculate the sum to infinity of this sequence 3. An athlete runs 20 km on a certain Monday. Thereafter, he increases the distance by \( 10 \% \) every day: Calculate the number of kilometres he ran:

Let's solve the problems step by step. ### Problem 2: Infinite Geometric Sequence **Given:** The first three terms of the geometric sequence are \( a_1 = 16 \), \( a_2 = 8 \), and \( a_3 = 4 \). #### Part a: Determine the \( n^{\text{th}} \) term of the sequence. 1. **Identify the common ratio \( r \)**: \[ r = \frac{a_2}{a_1} = \frac{8}{16} = \frac{1}{2} \] 2. **General formula for the \( n^{\text{th}} \) term**: The \( n^{\text{th}} \) term of a geometric sequence can be expressed as: \[ a_n = a_1 \cdot r^{n-1} \] 3. **Substituting the values**: \[ a_n = 16 \cdot \left(\frac{1}{2}\right)^{n-1} \] Thus, the \( n^{\text{th}} \) term of the sequence is: \[ a_n = 16 \cdot \left(\frac{1}{2}\right)^{n-1} \] #### Part b: Determine all possible values of \( n \) for which the sum of the first \( n \) terms is greater than 31. 1. **Sum of the first \( n \) terms \( S_n \)**: The formula for the sum of the first \( n \) terms of a geometric sequence is: \[ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \] 2. **Substituting the values**: \[ S_n = 16 \cdot \frac{1 - \left(\frac{1}{2}\right)^n}{1 - \frac{1}{2}} = 16 \cdot \frac{1 - \left(\frac{1}{2}\right)^n}{\frac{1}{2}} = 32 \cdot \left(1 - \left(\frac{1}{2}\right)^n\right) \] 3. **Setting up the inequality**: \[ 32 \cdot \left(1 - \left(\frac{1}{2}\right)^n\right) > 31 \] 4. **Solving the inequality**: \[ 1 - \left(\frac{1}{2}\right)^n > \frac{31}{32} \] \[ -\left(\frac{1}{2}\right)^n > -\frac{1}{32} \] \[ \left(\frac{1}{2}\right)^n < \frac{1}{32} \] 5. **Taking logarithm**: \[ n > 5 \] Thus, the possible values of \( n \) are \( n = 6, 7, 8, \ldots \) #### Part c: Calculate the sum to infinity of this sequence. 1. **Sum to infinity \( S_{\infty} \)**: The formula for the sum to infinity of a geometric sequence is: \[ S_{\infty} = \frac{a_1}{1 - r} \] 2. **Substituting the values**: \[ S_{\infty} = \frac{16}{1 - \frac{1}{2}} = \frac{16}{\frac{1}{2}} = 32 \] Thus, the sum to infinity of this sequence is: \[ S_{\infty} = 32 \] --- ### Problem 3: Athlete's Running Distance **Given:** The athlete runs 20 km on a certain Monday and increases the distance by \( 10\% \) every day. 1. **Initial distance**: \[ d_1 = 20 \text{ km} \] 2. **Distance on the \( n^{\text{th}} \) day**: The distance run on the \( n^{\text{th}} \) day can be expressed as: \[ d_n = d_1 \cdot (1 + 0.1)^{n-1} = 20 \cdot (1.1)^{n-1} \] 3. **Total distance run after \( n \) days**: The total distance run after \( n \) days is the sum of a geometric series: \[ D_n = d_1 \cdot \frac{(1 + r)^n - 1}{r} = 20 \cdot \frac{(1.1)^n - 1}{0.1} \] Thus, the total distance run after \( n \) days is: \[ D_n = 200 \cdot ((1.1)^n - 1) \] If you need a specific number of days or total distance, please provide that information!