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. 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:

Question

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. 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:

UpStudy AI · Accepted Answer

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!
**Problem 2: Infinite Geometric Sequence** - **Part a:** The $$ n^{\text{th}} $$ term is $$ a_n = 16 \times \left(\frac{1}{2}\right)^{n-1} $$. - **Part b:** The sum of the first $$ n $$ terms is greater than 31 when $$ n $$ is 6 or more. - **Part c:** The sum to infinity of the sequence is 32. **Problem 3: Athlete's Running Distance** The total distance run after $$ n $$ days is $$ D_n = 200 \times (1.1)^n - 200 $$ kilometers.

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Pre Algebra Questions