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:
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:
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].
