4) The sum of the lirst 5 ferms of an arithmetic sequence is 0 and the fifth term is 8 . Determin contmond diference and the first term. 5) In a geonetific series the first term is 36 , the last term is 16 and the sum of the series is 76 . Determine the coinmon ratio and the number of terns in the sequence.

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## **Problem 4**

**Given:**
- An arithmetic sequence.
- Sum of the first 5 terms is **0**.
- The fifth term is **8**.

**Find:**
- The **first term** $$ a $$.
- The **common difference** $$ d $$.

**Solution:**

For an arithmetic sequence, the $$ n $$-th term is given by:
$$ a_n = a + (n-1)d $$

1. **Fifth Term:**
   $$ a_5 = a + 4d = 8 $$
   
2. **Sum of the First 5 Terms:**
   The sum $$ S_5 $$ of the first 5 terms is:
   $$ S_5 = \frac{5}{2} (2a + 4d) = 0 $$
   Simplifying:
   $$ 5a + 10d = 0 $$
   $$ a + 2d = 0 $$
   $$ a = -2d $$
   
3. **Substitute $$ a $$ into the Fifth Term Equation:**
   $$ -2d + 4d = 8 $$
   $$ 2d = 8 $$
   $$ d = 4 $$
   
4. **Find $$ a $$:**
   $$ a = -2 \times 4 = -8 $$

**Answer:**
- **First term ($$ a $$)**: $$-8$$
- **Common difference ($$ d $$)**: $$4$$

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## **Problem 5**

**Given:**
- A geometric series.
- First term $$ a = 36 $$.
- Last term $$ a_n = 16 $$.
- Sum of the series $$ S_n = 76 $$.

**Find:**
- The **common ratio** $$ r $$.
- The **number of terms** $$ n $$.

**Solution:**

For a geometric series:
1. **$$ n $$-th Term:**
   $$ a_n = a \cdot r^{n-1} $$
   $$ 16 = 36 \cdot r^{n-1} $$
   $$ r^{n-1} = \frac{16}{36} = \frac{4}{9} $$
   $$ r^{n-1} = \left(\frac{2}{3}\right)^2 $$
   This suggests:
   $$ r = \frac{2}{3} $$
   $$ n - 1 = 2 $$
   $$ n = 3 $$
   
2. **Sum of the First $$ n $$ Terms:**
   $$ S_n = a \cdot \frac{1 - r^n}{1 - r} $$
   $$ 76 = 36 \cdot \frac{1 - \left(\frac{2}{3}\right)^3}{1 - \frac{2}{3}} $$
   $$ 76 = 36 \cdot \frac{1 - \frac{8}{27}}{\frac{1}{3}} $$
   $$ 76 = 36 \cdot \frac{\frac{19}{27}}{\frac{1}{3}} $$
   $$ 76 = 36 \cdot \frac{19}{9} $$
   $$ 76 = 76 $$
   
   The values are consistent.

**Answer:**
- **Common ratio ($$ r $$)**: $$\frac{2}{3}$$
- **Number of terms ($$ n $$)**: $$3$$

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**Problem 4:** - **First term ($$ a $$)**: $$-8$$ - **Common difference ($$ d $$)**: $$4$$ **Problem 5:** - **Common ratio ($$ r $$)**: $$\frac{2}{3}$$ - **Number of terms ($$ n $$)**: $$3$$

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