2. The first three terms of an inginite geometric sequence are 16,8 and 4 respectively. a) Determine the $$ n^{ ext {th }} $$ 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:

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2. The first three terms of an inginite geometric sequence are 16,8 and 4 respectively. a) Determine the $$ n^{	ext {th }} $$ 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:

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### 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!
**Problem 2: Geometric Sequence** a) The $$ n^{\text{th}} $$ term is $$ a_n = 16 \cdot \left(\frac{1}{2}\right)^{n-1} $$. b) The sum of the first $$ n $$ terms is greater than 31 when $$ n > 5 $$. c) The sum to infinity is 32. **Problem 3: Athlete's Running Distance** The distance on the $$ n^{\text{th}} $$ day is $$ d_n = 20 \cdot (1.1)^{n-1} $$.

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