(h) Consider the arithmetic series $$ \frac{3}{2}+2+\frac{5}{2}+\ldots $$ (1) Calculate $$ S_{12} $$. (2) Calculate the value of $$ k $$ if $$ S_{k}=231 $$. (3) Calculate the greatest number of terms that can be added, if the answer must be less than 300 .

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Let's analyze the given arithmetic series:

$$
\frac{3}{2},\ 2,\ \frac{5}{2},\ \ldots
$$

**Step 1: Identify the First Term and Common Difference**

- **First term ($$a_1$$)**: $$\frac{3}{2}$$
- **Second term ($$a_2$$)**: $$2 = \frac{4}{2}$$
  
The common difference ($$d$$) is:

$$
d = a_2 - a_1 = \frac{4}{2} - \frac{3}{2} = \frac{1}{2}
$$

So, the general term of the series is:

$$
a_n = a_1 + (n-1)d = \frac{3}{2} + (n-1) \cdot \frac{1}{2} = \frac{n+2}{2}
$$

### Part (1): Calculate $$ S_{12} $$

The sum of the first $$ n $$ terms of an arithmetic series is given by:

$$
S_n = \frac{n}{2} \left(2a_1 + (n-1)d\right)
$$

Substituting $$ n = 12 $$:

$$
S_{12} = \frac{12}{2} \left(2 \cdot \frac{3}{2} + 11 \cdot \frac{1}{2}\right) = 6 \left(3 + \frac{11}{2}\right) = 6 \cdot \frac{17}{2} = 51
$$

**Answer:** $$ S_{12} = 51 $$

### Part (2): Find $$ k $$ such that $$ S_k = 231 $$

Using the sum formula:

$$
\frac{k}{2} \left(2 \cdot \frac{3}{2} + (k-1) \cdot \frac{1}{2}\right) = 231
$$

Simplifying:

$$
\frac{k}{2} \left(3 + \frac{k-1}{2}\right) = 231 \
\frac{k(k+5)}{4} = 231 \
k(k + 5) = 924 \
k^2 + 5k - 924 = 0
$$

Solving the quadratic equation:

$$
k = \frac{-5 \pm \sqrt{25 + 3696}}{2} = \frac{-5 \pm 61}{2}
$$

Taking the positive root:

$$
k = \frac{56}{2} = 28
$$

**Answer:** $$ k = 28 $$

### Part (3): Find the Greatest Number of Terms with $$ S_n < 300 $$

We want the largest $$ n $$ such that:

$$
S_n = \frac{n(n + 5)}{4} < 300 \
n(n + 5) < 1200 \
n^2 + 5n - 1200 < 0
$$

Solving the quadratic inequality:

$$
n \approx \frac{-5 + \sqrt{4825}}{2} \approx \frac{-5 + 69.5}{2} \approx 32.25
$$

Since $$ n $$ must be an integer:

$$
n \leq 32
$$

**Verification:**

- For $$ n = 32 $$:

$$
S_{32} = \frac{32 \cdot 37}{4} = 296 < 300
$$

- For $$ n = 33 $$:

$$
S_{33} = \frac{33 \cdot 38}{4} = 313.5 > 300
$$

**Answer:** The greatest number of terms is **32**.

### Summary of Answers

1. $$ S_{12} = 51 $$
2. $$ k = 28 $$
3. The greatest number of terms is **32**.
**Answers:** 1. $$ S_{12} = 51 $$ 2. $$ k = 28 $$ 3. The greatest number of terms is **32**.

(h) Consider the arithmetic series
(1) Calculate .
(2) Calculate the value of if .
(3) Calculate the greatest number of terms that can be added, if the answer must
be less than 300 .

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(h) Consider the arithmetic series (1) Calculate . (2) Calculate the value of if . (3) Calculate the greatest number of terms that can be added, if the answer must be less than 300 .