Sure! We will analyze and solve the given series [tex]\((3+1)+(9+2)+(27+3)+\dots\)[/tex] up to the first 10 terms.
### Step-by-Step Solution:
1. Identify the Pattern:
Observe the first few terms of the series:
- The first term is [tex]\(3 + 1 = 4\)[/tex].
- The second term is [tex]\(9 + 2 = 11\)[/tex].
- The third term is [tex]\(27 + 3 = 30\)[/tex].
The terms can be represented in the form [tex]\(3^n + n\)[/tex], where [tex]\(n\)[/tex] starts from 1.
2. First Three Terms:
- The first term: [tex]\(3^1 + 1 = 3 + 1 = 4\)[/tex].
- The second term: [tex]\(3^2 + 2 = 9 + 2 = 11\)[/tex].
- The third term: [tex]\(3^3 + 3 = 27 + 3 = 30\)[/tex].
3. Summing Up to the 10th Term:
We will sum the first 10 terms of the series [tex]\(3^n + n\)[/tex].
4. Sum Calculation:
Let us denote the sum of the series up to the 10th term as [tex]\(S_{10}\)[/tex].
[tex]\[
S_{10} = \sum_{n=1}^{10} (3^n + n)
\][/tex]
We add each term individually from [tex]\(n = 1\)[/tex] to [tex]\(n = 10\)[/tex]:
[tex]\[
S_{10} = (3^1 + 1) + (3^2 + 2) + (3^3 + 3) + \ldots + (3^{10} + 10)
\][/tex]
Calculating the sum of these terms:
[tex]\[
S_{10} = 4 + 11 + 30 + \cdots + 3^{10} + 10
\][/tex]
5. Final Result:
The sum [tex]\(S_{10}\)[/tex] turns out to be 88627.
### Conclusion:
- The first term is [tex]\(4\)[/tex].
- The second term is [tex]\(11\)[/tex].
- The third term is [tex]\(30\)[/tex].
- The sum of the first 10 terms of the series is [tex]\(88627\)[/tex].
Just to recap, our results for the given question are:
- First term: [tex]\(4\)[/tex]
- Second term: [tex]\(11\)[/tex]
- Third term: [tex]\(30\)[/tex]
- Sum of the first 10 terms: [tex]\(88627\)[/tex]
