QUESTION 2 2.1 The arithmetic series $$ 3+5+7+\ldots+103 $$ is given. 2.1.1 Determine the general term $$ 2.1 .2 \quad $$ Calculate the number of terms in the given series $$ 2.1 .3 \quad $$ Hence, write the series in sigma notation 2.2 Calculate $$ \boldsymbol{n} $$ if : 2.3 $$ \sum_{n=2}^{n}(2 \boldsymbol{n}+\mathbf{1})\mathbf{1 9 2} $$ The sum of the $$ 1^{ ext {st }} $$ and $$ 2^{ ext {nd }} $$ terms of a geometric series is 30 , and the sum to infinity is 54 . Determine the $$ 1^{ ext {st }} $$ term $$ \boldsymbol{a} $$, and the common ratio, $$ \boldsymbol{r} $$, with $$ \boldsymbol{r}\mathbf{0} $$.

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QUESTION 2 2.1 The arithmetic series $$ 3+5+7+\ldots+103 $$ is given. 2.1.1 Determine the general term $$ 2.1 .2 \quad $$ Calculate the number of terms in the given series $$ 2.1 .3 \quad $$ Hence, write the series in sigma notation 2.2 Calculate $$ \boldsymbol{n} $$ if : 2.3 $$ \sum_{n=2}^{n}(2 \boldsymbol{n}+\mathbf{1})>\mathbf{1 9 2} $$ The sum of the $$ 1^{	ext {st }} $$ and $$ 2^{	ext {nd }} $$ terms of a geometric series is 30 , and the sum to infinity is 54 . Determine the $$ 1^{	ext {st }} $$ term $$ \boldsymbol{a} $$, and the common ratio, $$ \boldsymbol{r} $$, with $$ \boldsymbol{r}>\mathbf{0} $$.

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### 2.1 Arithmetic Series $$ 3 + 5 + 7 + \ldots + 103 $$

#### 2.1.1 Determine the general term

The series is an arithmetic series where the first term $$ a = 3 $$ and the common difference $$ d = 5 - 3 = 2 $$.

The general term $$ T_n $$ of an arithmetic series can be expressed as:
$$
T_n = a + (n - 1)d
$$
Substituting the values of $$ a $$ and $$ d $$:
$$
T_n = 3 + (n - 1) \cdot 2 = 3 + 2n - 2 = 2n + 1
$$

#### 2.1.2 Calculate the number of terms in the given series

To find the number of terms $$ n $$, we set the last term equal to 103:
$$
T_n = 103
$$
Substituting the expression for $$ T_n $$:
$$
2n + 1 = 103
$$
Now, solve for $$ n $$:
$$
2n = 103 - 1 = 102 \
n = \frac{102}{2} = 51
$$

#### 2.1.3 Write the series in sigma notation

The series can be expressed in sigma notation as:
$$
\sum_{n=1}^{51} (2n + 1)
$$

### 2.2 Calculate $$ n $$ if $$ \sum_{n=2}^{n}(2n + 1) > 192 $$

First, we need to calculate the sum $$ \sum_{n=2}^{k} (2n + 1) $$ for some $$ k $$.

The sum can be calculated as follows:
$$
\sum_{n=2}^{k} (2n + 1) = \sum_{n=2}^{k} 2n + \sum_{n=2}^{k} 1
$$
The first part is:
$$
\sum_{n=2}^{k} 2n = 2 \sum_{n=2}^{k} n = 2 \left( \frac{k(k + 1)}{2} - 1 \right) = k(k + 1) - 2
$$
The second part is:
$$
\sum_{n=2}^{k} 1 = k - 1
$$
Combining these:
$$
\sum_{n=2}^{k} (2n + 1) = k(k + 1) - 2 + (k - 1) = k^2 + k - 3
$$
Now, we set up the inequality:
$$
k^2 + k - 3 > 192
$$
This simplifies to:
$$
k^2 + k - 195 > 0
$$
Now, we will solve the quadratic equation $$ k^2 + k - 195 = 0 $$ using the quadratic formula:
$$
k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1 + 780}}{2} = \frac{-1 \pm \sqrt{781}}{2}
$$
Calculating $$ \sqrt{781} $$:
$$
\sqrt{781} \approx 27.93
$$
Thus:
$$
k \approx \frac{-1 + 27.93}{2} \approx 13.465
$$
Since $$ k $$ must be an integer, we check $$ k = 14 $$ and $$ k = 13 $$.

Calculating for $$ k = 14 $$:
$$
14^2 + 14 - 195 = 196 + 14 - 195 = 15 > 0
$$
Calculating for $$ k = 13 $$:
$$
13^2 + 13 - 195 = 169 + 13 - 195 = -13 < 0
$$
Thus, $$ k = 14 $$ is the smallest integer satisfying the inequality.

### 2.3 Geometric Series

The first term $$ a $$ and the common ratio $$ r $$ satisfy the following conditions:
1. $$ a + ar = 30 $$
2. $$ \frac{a}{1 - r} = 54 $$

From the first equation:
$$
a(1 + r) = 30 \quad \text{(1)}
$$
From the second equation:
$$
a = 54(1 - r) \quad \text{(2)}
$$
Substituting (2) into (1):
$$
54(1 - r)(1 + r) = 30
$$
Expanding:
$$
54(1 - r^2) = 30
$$
This simplifies to:
$$
54 - 54r^2 = 30 \
54r^2 = 54 - 30 = 24 \
r^2 = \frac{24}{54} = \frac{4}{9} \
r = \frac{2}{3} \quad \text{(since $$ r > 0 $$)}
$$
Now substituting $$ r $$ back into (2) to find $$ a $$:
$$
a = 54(1 - \frac{2}{3}) = 54 \cdot \frac{1}{3} = 18
$$

### Final Answers
- General term: $$ 2n + 1 $$
- Number of terms: $$ 51 $$
- Sigma notation: $$ \sum_{n=1}^{51} (2n + 1) $$
- $$ n $$: $$ 14 $$
- First term $$ a $$: $$ 18 $$
- Common ratio $$ r $$: $$ \frac{2}{3} $$
- General term: $$ 2n + 1 $$ - Number of terms: 51 - Sigma notation: $$ \sum_{n=1}^{51} (2n + 1) $$ - $$ n $$: 14 - First term $$ a $$: 18 - Common ratio $$ r $$: $$ \frac{2}{3} $$

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