Sure, let's break this down step by step!
To find the sum of the series [tex]\(1 + 11 + 101 + 1001 + \ldots\)[/tex] up to [tex]\(n\)[/tex] terms, let's first observe the pattern in the series.
1. Identify the pattern:
- The 1st term is [tex]\(1\)[/tex].
- The 2nd term is [tex]\(11\)[/tex].
- The 3rd term is [tex]\(101\)[/tex].
- The 4th term is [tex]\(1001\)[/tex].
Each term can be represented as a sum of powers of 10.
2. Express each term:
- 1st term: [tex]\(1\)[/tex]
- 2nd term: [tex]\(10 + 1 = 11\)[/tex]
- 3rd term: [tex]\(100 + 1 = 101\)[/tex]
- 4th term: [tex]\(1000 + 1 = 1001\)[/tex]
Let's generalize a term in this series:
- The [tex]\(i\)[/tex]-th term can be expressed as the sum of different powers of 10:
[tex]\[ \text{Term}(i) = 10^{i-1} + 10^{i-2} + \cdots + 10^1 + 10^0 \][/tex]
3. Calculate each term and the sum:
- For [tex]\(n = 10\)[/tex], we'll calculate the first 10 terms:
- Term 1: [tex]\(1\)[/tex]
- Term 2: [tex]\(11\)[/tex]
- Term 3: [tex]\(101\)[/tex]
- Term 4: [tex]\(1001\)[/tex]
- Term 5: [tex]\(10001\)[/tex]
- Term 6: [tex]\(100001\)[/tex]
- Term 7: [tex]\(1000001\)[/tex]
- Term 8: [tex]\(10000001\)[/tex]
- Term 9: [tex]\(100000001\)[/tex]
- Term 10: [tex]\(1000000001\)[/tex]
4. Sum up the terms:
To find the sum to [tex]\(n\)[/tex] terms, we add up these terms:
[tex]\[
\begin{align*}
\text{Sum}(10) & = 1 + 11 + 101 + 1001 + 10001 + 100001 + 1000001 + 10000001 + 100000001 + 1000000001 \\
& = 1234567900
\end{align*}
\][/tex]
Additionally, the last term (10th term) is:
[tex]\[ 1000000001 \][/tex]
So, for [tex]\(n = 10\)[/tex]:
- The 10th term in the series is [tex]\(1111111111\)[/tex].
- The sum of the first 10 terms of the series is [tex]\(1234567900\)[/tex].
Therefore, the results are:
- The 10th term is [tex]\(1111111111\)[/tex].
- The sum of the first 10 terms is [tex]\(1234567900\)[/tex].
