Determine the sum to 50 terms of the arithmetic series of which the 7 th term is 33 and the 15 th term is 65 . In an arithmetic series, $$ T_{3}=-8 $$ and $$ T_{12}=-35 $$. Determine the number of terms in the series if its sum is -100 . In an arithmetic series, $$ T_{7}=45 $$ and $$ T_{3}=17 $$. Determine the number of terms in the series if $$ S_{n}=123 $$. The sum of the first 6 terms of an arithmetic series is 90 and the last term (6th) is 25 . Determine the first term and the common difference.

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### **1. Sum of the First 50 Terms**

**Given:**
- 7th term, $$ T_7 = 33 $$
- 15th term, $$ T_{15} = 65 $$

**Find:** Sum of the first 50 terms, $$ S_{50} $$.

**Solution:**

1. **Find the First Term ($$ a $$) and Common Difference ($$ d $$):**
   $$
   T_n = a + (n - 1)d
   $$
   $$
   T_7 = a + 6d = 33 \quad \text{(1)}
   $$
   $$
   T_{15} = a + 14d = 65 \quad \text{(2)}
   $$
   
   Subtract equation (1) from (2):
   $$
   (a + 14d) - (a + 6d) = 65 - 33
   $$
   $$
   8d = 32 \Rightarrow d = 4
   $$
   
   Substitute $$ d = 4 $$ into equation (1):
   $$
   a + 6(4) = 33 \Rightarrow a = 9
   $$

2. **Calculate the Sum $$ S_{50} $$:**
   $$
   S_n = \frac{n}{2} [2a + (n - 1)d]
   $$
   $$
   S_{50} = \frac{50}{2} [2(9) + 49(4)] = 25 [18 + 196] = 25 \times 214 = 5350
   $$

**Answer:** The sum of the first 50 terms is **5350**.

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### **2. Number of Terms with $$ S_n = -100 $$**

**Given:**
- 3rd term, $$ T_3 = -8 $$
- 12th term, $$ T_{12} = -35 $$
- Sum of the series, $$ S_n = -100 $$

**Find:** Number of terms, $$ n $$.

**Solution:**

1. **Find $$ a $$ and $$ d $$:**
   $$
   T_3 = a + 2d = -8 \quad \text{(1)}
   $$
   $$
   T_{12} = a + 11d = -35 \quad \text{(2)}
   $$
   
   Subtract equation (1) from (2):
   $$
   9d = -27 \Rightarrow d = -3
   $$
   
   Substitute $$ d = -3 $$ into equation (1):
   $$
   a + 2(-3) = -8 \Rightarrow a = -2
   $$

2. **Find $$ n $$ Using the Sum Formula:**
   $$
   S_n = \frac{n}{2} [2a + (n - 1)d] = -100
   $$
   $$
   \frac{n}{2} [2(-2) + (n - 1)(-3)] = -100
   $$
   $$
   \frac{n}{2} [-4 - 3n + 3] = -100
   $$
   $$
   \frac{n}{2} (-3n - 1) = -100
   $$
   $$
   -3n^2 - n = -200
   $$
   $$
   3n^2 + n - 200 = 0
   $$
   
   Solve the quadratic equation:
   $$
   n = \frac{-1 \pm \sqrt{1 + 2400}}{6} = \frac{-1 \pm 49}{6}
   $$
   $$
   n = 8 \quad (\text{since } n \text{ must be positive})
   $$

**Answer:** The series has **8 terms**.

---

### **3. Number of Terms with $$ S_n = 123 $$**

**Given:**
- 7th term, $$ T_7 = 45 $$
- 3rd term, $$ T_3 = 17 $$
- Sum of the series, $$ S_n = 123 $$

**Find:** Number of terms, $$ n $$.

**Solution:**

1. **Find $$ a $$ and $$ d $$:**
   $$
   T_7 = a + 6d = 45 \quad \text{(1)}
   $$
   $$
   T_3 = a + 2d = 17 \quad \text{(2)}
   $$
   
   Subtract equation (2) from (1):
   $$
   4d = 28 \Rightarrow d = 7
   $$
   
   Substitute $$ d = 7 $$ into equation (2):
   $$
   a + 2(7) = 17 \Rightarrow a = 3
   $$

2. **Find $$ n $$ Using the Sum Formula:**
   $$
   S_n = \frac{n}{2} [2a + (n - 1)d] = 123
   $$
   $$
   \frac{n}{2} [6 + 7(n - 1)] = 123
   $$
   $$
   \frac{n}{2} [7n - 1] = 123
   $$
   $$
   7n^2 - n - 246 = 0
   $$
   
   Solve the quadratic equation:
   $$
   n = \frac{1 \pm \sqrt{1 + 6864}}{14} = \frac{1 \pm 83}{14}
   $$
   $$
   n = 6 \quad (\text{since } n \text{ must be positive})
   $$

**Answer:** The series has **6 terms**.

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### **4. First Term and Common Difference**

**Given:**
- Sum of the first 6 terms, $$ S_6 = 90 $$
- 6th term, $$ T_6 = 25 $$

**Find:** First term ($$ a $$) and common difference ($$ d $$).

**Solution:**

1. **Set Up Equations:**
   $$
   T_6 = a + 5d = 25 \quad \text{(1)}
   $$
   $$
   S_6 = \frac{6}{2} [2a + 5d] = 90
   $$
   $$
   3(2a + 5d) = 90 \Rightarrow 2a + 5d = 30 \quad \text{(2)}
   $$

2. **Solve the System of Equations:**
   Subtract equation (1) from equation (2):
   $$
   (2a + 5d) - (a + 5d) = 30 - 25
   $$
   $$
   a = 5
   $$
   
   Substitute $$ a = 5 $$ into equation (1):
   $$
   5 + 5d = 25 \Rightarrow d = 4
   $$

**Answer:** The first term is **5** and the common difference is **4**.

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If you have any further questions or need additional clarification, feel free to ask!
**1. Sum of the First 50 Terms:** - First term ($$ a $$) = 9 - Common difference ($$ d $$) = 4 - Sum ($$ S_{50} $$) = 5350 **2. Number of Terms with $$ S_n = -100 $$:** - First term ($$ a $$) = -2 - Common difference ($$ d $$) = -3 - Number of terms ($$ n $$) = 8 **3. Number of Terms with $$ S_n = 123 $$:** - First term ($$ a $$) = 3 - Common difference ($$ d $$) = 7 - Number of terms ($$ n $$) = 6 **4. First Term and Common Difference:** - First term ($$ a $$) = 5 - Common difference ($$ d $$) = 4

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