To find the sum of the sequence [tex]$1 + 11 + 2 + 12 + 3 + 13 + \ldots + 19 + 29$[/tex], we need to carefully examine and break down the sequence into two distinct arithmetic sequences, then sum those sequences and add their results.
### Step 1: Identify the sequences
The given sequence can be split into two different arithmetic sequences as follows:
1. First arithmetic sequence (S1): 1, 2, 3, ..., 19
2. Second arithmetic sequence (S2): 11, 12, 13, ..., 29
### Step 2: Sum of the first arithmetic sequence (S1)
1. Identify the first term (a1) and the last term (l1):
- The first term for S1: [tex]\( a1 = 1 \)[/tex]
- The last term for S1: [tex]\( l1 = 19 \)[/tex]
2. Number of terms (n1) in the first sequence:
- Number of terms, n1: [tex]\( n1 = 19 \)[/tex]
3. Sum of the first sequence (S1) using the formula for the sum of an arithmetic sequence [tex]\( \text{Sum} = \frac{n}{2} \cdot (a + l) \)[/tex]:
[tex]\[
\text{Sum}_1 = \frac{19}{2} \times (1 + 19) = 19 \times 10 = 190
\][/tex]
### Step 3: Sum of the second arithmetic sequence (S2)
1. Identify the first term (a2) and the last term (l2):
- The first term for S2: [tex]\( a2 = 11 \)[/tex]
- The last term for S2: [tex]\( l2 = 29 \)[/tex]
2. Number of terms (n2) in the second sequence:
- Number of terms, n2: [tex]\( l2 - a2 + 1 = 29 - 11 + 1 = 19 \)[/tex]
3. Sum of the second sequence (S2) using the same formula:
[tex]\[
\text{Sum}_2 = \frac{19}{2} \times (11 + 29) = 19 \times 20 = 380
\][/tex]
### Step 4: Combine the results
To find the total sum of the given sequence, we add the sums of the two sequences:
[tex]\[
\text{Total Sum} = \text{Sum}_1 + \text{Sum}_2 = 190 + 380 = 570
\][/tex]
Therefore, the sum of the sequence [tex]\(1 + 11 + 2 + 12 + 3 + 13 + \ldots + 19 + 29\)[/tex] is [tex]\( 570 \)[/tex].
