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. (2) 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:

Let's solve the problems step by step. ### Problem 2: Geometric Sequence **Given:** The first three terms of the geometric sequence are \( a_1 = 16 \), \( a_2 = 8 \), and \( a_3 = 4 \). #### a) Determine the \( n^{\text{th}} \) term of the sequence. In a geometric sequence, the \( n^{\text{th}} \) term can be expressed as: \[ a_n = a_1 \cdot r^{n-1} \] where \( r \) is the common ratio. To find \( r \), we can use the first two terms: \[ r = \frac{a_2}{a_1} = \frac{8}{16} = \frac{1}{2} \] Now, we can express the \( n^{\text{th}} \) term: \[ a_n = 16 \cdot \left(\frac{1}{2}\right)^{n-1} \] #### b) Determine all possible values of \( n \) for which the sum of the first \( n \) terms of this sequence is greater than 31. The sum of the first \( n \) terms of a geometric sequence is given by: \[ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \] Substituting the known values: \[ S_n = 16 \cdot \frac{1 - \left(\frac{1}{2}\right)^n}{1 - \frac{1}{2}} = 16 \cdot \frac{1 - \left(\frac{1}{2}\right)^n}{\frac{1}{2}} = 32 \cdot \left(1 - \left(\frac{1}{2}\right)^n\right) \] We need to find \( n \) such that: \[ 32 \cdot \left(1 - \left(\frac{1}{2}\right)^n\right) > 31 \] Dividing both sides by 32: \[ 1 - \left(\frac{1}{2}\right)^n > \frac{31}{32} \] Rearranging gives: \[ -\left(\frac{1}{2}\right)^n > -\frac{1}{32} \] \[ \left(\frac{1}{2}\right)^n < \frac{1}{32} \] Since \( \frac{1}{32} = \left(\frac{1}{2}\right)^5 \), we have: \[ n > 5 \] Thus, the possible values of \( n \) are \( n = 6, 7, 8, \ldots \) #### c) Calculate the sum to infinity of this sequence. The sum to infinity \( S \) of a geometric series is given by: \[ S = \frac{a_1}{1 - r} \] Substituting the known values: \[ S = \frac{16}{1 - \frac{1}{2}} = \frac{16}{\frac{1}{2}} = 32 \] ### Problem 3: Athlete's Running Distance **Given:** The athlete runs 20 km on a certain Monday and increases the distance by \( 10\% \) every day. The distance run on the \( n^{\text{th}} \) day can be expressed as: \[ d_n = 20 \cdot (1.1)^{n-1} \] If you have any specific questions or further calculations regarding this problem, please let me know!