Consider the sequence $$ \frac{1}{2} ; 4 ; \frac{1}{4} ; 7 ; \frac{1}{8} ; 10 ; \ldots \ldots $$ 5.6.1 If the pattern continues in the same way, write down the next TWO terms in the sequence 5.6.2 Calculate the sum of the first 50 terms of the sequence.

Question

UpStudy AI · Accepted Answer

To solve the problem, let's first analyze the given sequence:

$$
\frac{1}{2}, 4, \frac{1}{4}, 7, \frac{1}{8}, 10, \ldots
$$

### Step 1: Identify the Pattern

The sequence alternates between two types of terms:
1. **Fractional terms**: These appear at the 1st, 3rd, 5th, etc. positions.
2. **Whole number terms**: These appear at the 2nd, 4th, 6th, etc. positions.

#### Fractional Terms:
- The first term is $$ \frac{1}{2} $$ (1st position).
- The third term is $$ \frac{1}{4} $$ (3rd position).
- The fifth term is $$ \frac{1}{8} $$ (5th position).

The pattern for the fractional terms can be observed as:
$$
\frac{1}{2^n} \quad \text{where } n = 1, 2, 3, \ldots
$$
Thus, the next fractional term (7th position) will be:
$$
\frac{1}{2^4} = \frac{1}{16}
$$

#### Whole Number Terms:
- The second term is $$ 4 $$ (2nd position).
- The fourth term is $$ 7 $$ (4th position).
- The sixth term is $$ 10 $$ (6th position).

The whole number terms appear to increase by $$ 3 $$:
$$
4, 7, 10, \ldots
$$
This can be expressed as:
$$
3n + 1 \quad \text{where } n = 1, 2, 3, \ldots
$$
Thus, the next whole number term (8th position) will be:
$$
3 \cdot 4 + 1 = 13
$$

### Step 2: Write Down the Next Two Terms
From the analysis, the next two terms in the sequence are:
1. $$ \frac{1}{16} $$ (7th position)
2. $$ 13 $$ (8th position)

### Step 3: Calculate the Sum of the First 50 Terms

#### Fractional Terms:
The fractional terms are:
$$
\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots, \frac{1}{2^{25}} \quad \text{(25 terms)}
$$
This is a geometric series with the first term $$ a = \frac{1}{2} $$ and common ratio $$ r = \frac{1}{2} $$.

The sum $$ S $$ of the first $$ n $$ terms of a geometric series is given by:
$$
S_n = a \frac{1 - r^n}{1 - r}
$$
For our series:
$$
S_{25} = \frac{1}{2} \frac{1 - \left(\frac{1}{2}\right)^{25}}{1 - \frac{1}{2}} = \frac{1}{2} \cdot 2 \left(1 - \frac{1}{2^{25}}\right) = 1 - \frac{1}{2^{25}}
$$

#### Whole Number Terms:
The whole number terms are:
$$
4, 7, 10, \ldots, 148 \quad \text{(25 terms)}
$$
This is an arithmetic series where:
- First term $$ a = 4 $$
- Last term $$ l = 148 $$
- Number of terms $$ n = 25 $$

The sum $$ S $$ of an arithmetic series is given by:
$$
S_n = \frac{n}{2} (a + l)
$$
Thus:
$$
S_{25} = \frac{25}{2} (4 + 148) = \frac{25}{2} \cdot 152 = 25 \cdot 76 = 1900
$$

### Step 4: Total Sum of the First 50 Terms
Now, we can calculate the total sum of the first 50 terms:
$$
\text{Total Sum} = S_{\text{fractional}} + S_{\text{whole number}} = \left(1 - \frac{1}{2^{25}}\right) + 1900
$$

Now, let's calculate the total sum:
$$
\text{Total Sum} \approx 1 + 1900 = 1901 \quad \text{(since } \frac{1}{2^{25}} \text{ is very small)}
$$

### Final Answers
5.6.1 The next two terms in the sequence are:
- $$ \frac{1}{16} $$
- $$ 13 $$

5.6.2 The sum of the first 50 terms of the sequence is approximately:
- $$ 1901 $$
The next two terms in the sequence are $$ \frac{1}{16} $$ and $$ 13 $$. The sum of the first 50 terms is approximately 1901.

Consider the sequence \( \frac{1}{2} ; 4 ; \frac{1}{4} ; 7 ; \frac{1}{8} ; 10 ; \ldots \ldots \) 5.6.1 If the pattern continues in the same way, write down the next TWO terms in the sequence 5.6.2 Calculate the sum of the first 50 terms of the sequence.

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Pre Algebra Questions