Let's solve each sum step by step.
### First Sequence: [tex]\( 9 + 4 + (-1) + \ldots + (-556) \)[/tex]
The sequence is in arithmetic progression where:
- The first term [tex]\( a_1 = 9 \)[/tex]
- The common difference [tex]\( d = 4 - 9 = -5 \)[/tex]
- The last term [tex]\( a_n = -556 \)[/tex]
To find the number of terms [tex]\( n \)[/tex] required for the sequence to reach [tex]\(-556\)[/tex], we use the formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence:
[tex]\[ a_n = a_1 + (n - 1)d \][/tex]
Plugging in the values:
[tex]\[ -556 = 9 + (n - 1)(-5) \][/tex]
Simplify and solve for [tex]\( n \)[/tex]:
[tex]\[ -556 = 9 - 5n + 5 \][/tex]
[tex]\[ -556 = 14 - 5n \][/tex]
[tex]\[ -570 = -5n \][/tex]
[tex]\[ n = 114 \][/tex]
So, the number of terms [tex]\( n = 114 \)[/tex].
Now, we calculate the sum of the first sequence. The formula for the sum [tex]\( S_n \)[/tex] of an arithmetic sequence is:
[tex]\[ S_n = \frac{n}{2} (a_1 + a_n) \][/tex]
Plugging in the values:
[tex]\[ S_{114} = \frac{114}{2} (9 + (-556)) \][/tex]
[tex]\[ S_{114} = 57 \times (-547) \][/tex]
[tex]\[ S_{114} = -31179 \][/tex]
So, the sum of the first sequence is [tex]\( -31179 \)[/tex].
### Second Sequence: [tex]\( \sum_{i=1}^{81} (3i - 10) \)[/tex]
This sequence can be treated as the sum of an arithmetic sequence where the general term [tex]\( a_i = 3i - 10 \)[/tex].
First, we identify the first term [tex]\( a_1 \)[/tex] and the last term [tex]\( a_{81} \)[/tex]:
- The first term: [tex]\( a_1 = 3(1) - 10 = -7 \)[/tex]
- The last term: [tex]\( a_{81} = 3(81) - 10 = 243 - 10 = 233 \)[/tex]
The number of terms [tex]\( n = 81 \)[/tex].
Now, we calculate the sum using the arithmetic series sum formula:
[tex]\[ S_n = \frac{n}{2} (a_1 + a_n) \][/tex]
Plugging in the values:
[tex]\[ S_{81} = \frac{81}{2} (-7 + 233) \][/tex]
[tex]\[ S_{81} = \frac{81}{2} \times 226 \][/tex]
[tex]\[ S_{81} = 40.5 \times 226 \][/tex]
[tex]\[ S_{81} = 9153 \][/tex]
So, the sum of the second sequence is [tex]\( 9153 \)[/tex].
### Final Results
- The sum of [tex]\( 9 + 4 + (-1) + \ldots + (-556) \)[/tex] is [tex]\( -31179 \)[/tex].
- The sum of [tex]\( \sum_{i=1}^{81} (3i - 10) \)[/tex] is [tex]\( 9153 \)[/tex].
