Find the sum of:
24 terms and n terms of `2 (1)/(2) , 3 (1)/(3) , 4 (1)/(6) ,5 ,......,`

Answer

To find the sum of the first 24 terms and the sum of the first n terms of the sequence given by the terms \(2 \frac{1}{2}, 3 \frac{1}{3}, 4 \frac{1}{6}, 5, \ldots\), we will first convert the mixed fractions into improper fractions, identify the pattern, and then use the formula for the sum of an arithmetic progression (AP).

### Step-by-step Solution:

1. **Convert Mixed Numbers to Improper Fractions:**
- The first term is \(2 \frac{1}{2} = \frac{5}{2}\).
- The second term is \(3 \frac{1}{3} = \frac{10}{3}\).
- The third term is \(4 \frac{1}{6} = \frac{25}{6}\).
- The fourth term is \(5 = \frac{5}{1}\).

2. **Identify the Terms:**
- The terms are:
- \(a_1 = \frac{5}{2}\)
- \(a_2 = \frac{10}{3}\)
- \(a_3 = \frac{25}{6}\)
- \(a_4 = 5\)

3. **Calculate the Common Difference (d):**
- To find the common difference \(d\), we calculate:
\[ d = a_2 - a_1 = \frac{10}{3} - \frac{5}{2} \]
- Finding a common denominator (which is 6):
\[ d = \left(\frac{10 \times 2}{3 \times 2}\right) - \left(\frac{5 \times 3}{2 \times 3}\right) = \frac{20}{6} - \frac{15}{6} = \frac{5}{6} \]

4. **Use the Sum Formula for AP:**
- The formula for the sum of the first \(n\) terms of an AP is:
\[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \]
- Here, \(a = \frac{5}{2}\), \(d = \frac{5}{6}\), and \(n = 24\).

5. **Plug in the Values:**
- Calculate \(S_{24}\):
\[ S_{24} = \frac{24}{2} \left(2 \cdot \frac{5}{2} + (24-1) \cdot \frac{5}{6}\right) \]
- Simplifying:
\[ S_{24} = 12 \left(5 + 23 \cdot \frac{5}{6}\right) \]
- Calculate \(23 \cdot \frac{5}{6}\):
\[ 23 \cdot \frac{5}{6} = \frac{115}{6} \]
- Now, add:
\[ 5 = \frac{30}{6} \quad \Rightarrow \quad 5 + \frac{115}{6} = \frac{30 + 115}{6} = \frac{145}{6} \]
- Now, substitute back:
\[ S_{24} = 12 \cdot \frac{145}{6} = \frac{1740}{6} = 290 \]

6. **Conclusion:**
- Therefore, the sum of the first 24 terms is \(290\).