The mean of the first 10 terms : ` 7xx8 , 10xx10 , 13xx12 , . . . ,`

Answer

To find the mean of the first 10 terms of the given sequence: \( 7xx8, 10xx10, 13xx12, \ldots \), we first need to identify the pattern in the sequence and then calculate the sum of the first 10 terms.

### Step-by-Step Solution:

1. **Identify the Sequence**:
The terms can be expressed as:
- First term: \( 7 \)
- Second term: \( 10 \)
- Third term: \( 13 \)
- The pattern in the first part of the terms is an arithmetic progression (AP) with a common difference of \( 3 \).

The first terms are: \( 7, 10, 13, \ldots \)

2. **Find the General Formula for the First Part**:
The \( n \)-th term of the first part can be expressed as:
\[ a_n = 7 + (n - 1) \cdot 3 = 3n + 4 \]

3. **Identify the Second Part of the Terms**:
The second part of the terms is:
- First term: \( 8 \)
- Second term: \( 10 \)
- Third term: \( 12 \)
- The pattern in the second part is also an AP with a common difference of \( 2 \).

The second terms are: \( 8, 10, 12, \ldots \)

4. **Find the General Formula for the Second Part**:
The \( n \)-th term of the second part can be expressed as:
\[ b_n = 8 + (n - 1) \cdot 2 = 2n + 6 \]

5. **Combine the Two Parts**:
The \( n \)-th term of the overall sequence can be expressed as:
\[ T_n = a_n \cdot b_n = (3n + 4)(2n + 6) \]

6. **Expand the Expression**:
\[ T_n = (3n + 4)(2n + 6) = 6n^2 + 18n + 8n + 24 = 6n^2 + 26n + 24 \]

7. **Calculate the Sum of the First 10 Terms**:
We need to find:
\[ S_{10} = \sum_{n=1}^{10} T_n = \sum_{n=1}^{10} (6n^2 + 26n + 24) \]
This can be separated into three sums:
\[ S_{10} = 6 \sum_{n=1}^{10} n^2 + 26 \sum_{n=1}^{10} n + \sum_{n=1}^{10} 24 \]

8. **Use the Formulas for Summation**:
- The sum of the first \( n \) natural numbers:
\[ \sum_{k=1}^{n} k = \frac{n(n + 1)}{2} \]
- The sum of the squares of the first \( n \) natural numbers:
\[ \sum_{k=1}^{n} k^2 = \frac{n(n + 1)(2n + 1)}{6} \]

For \( n = 10 \):
\[ \sum_{k=1}^{10} k = \frac{10 \cdot 11}{2} = 55 \]
\[ \sum_{k=1}^{10} k^2 = \frac{10 \cdot 11 \cdot 21}{6} = 385 \]

9. **Calculate Each Part**:
\[ S_{10} = 6 \cdot 385 + 26 \cdot 55 + 24 \cdot 10 \]
\[ = 2310 + 1430 + 240 = 3980 \]

10. **Find the Mean**:
The mean is given by:
\[ \text{Mean} = \frac{S_{10}}{10} = \frac{3980}{10} = 398 \]

### Final Answer:
The mean of the first 10 terms is \( 398 \).