Let's find the sum of the first [tex]\( n \)[/tex] terms for each of the given arithmetic sequences. In an arithmetic sequence, the sum of the first [tex]\( n \)[/tex] terms ([tex]\( S_n \)[/tex]) is given by the formula:
[tex]\[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \][/tex]
where:
- [tex]\( a \)[/tex] is the first term,
- [tex]\( d \)[/tex] is the common difference between the terms,
- [tex]\( n \)[/tex] is the number of terms.
### 1. Find [tex]\( S_{20} \)[/tex] of the arithmetic sequence [tex]\( -8, 1, 10, 19, \ldots \)[/tex]
For this sequence:
- The first term ([tex]\( a \)[/tex]) is [tex]\( -8 \)[/tex].
- The common difference ([tex]\( d \)[/tex]) is [tex]\( 1 - (-8) = 9 \)[/tex].
- The number of terms ([tex]\( n \)[/tex]) is [tex]\( 20 \)[/tex].
Using the formula:
[tex]\[ S_{20} = \frac{20}{2} \left(2(-8) + (20-1) \cdot 9\right) \][/tex]
[tex]\[ S_{20} = 10 \left(-16 + 171\right) \][/tex]
[tex]\[ S_{20} = 10 \cdot 155 \][/tex]
[tex]\[ S_{20} = 1550 \][/tex]
Thus, [tex]\( S_{20} = 1550. \)[/tex]
### 2. Find [tex]\( S_{30} \)[/tex] of the arithmetic sequence [tex]\( 4, 10, 16, 22, \ldots \)[/tex]
For this sequence:
- The first term ([tex]\( a \)[/tex]) is [tex]\( 4 \)[/tex].
- The common difference ([tex]\( d \)[/tex]) is [tex]\( 10 - 4 = 6 \)[/tex].
- The number of terms ([tex]\( n \)[/tex]) is [tex]\( 30 \)[/tex].
Using the formula:
[tex]\[ S_{30} = \frac{30}{2} \left(2 \cdot 4 + (30-1) \cdot 6\right) \][/tex]
[tex]\[ S_{30} = 15 \left(8 + 174\right) \][/tex]
[tex]\[ S_{30} = 15 \cdot 182 \][/tex]
[tex]\[ S_{30} = 2730 \][/tex]
Thus, [tex]\( S_{30} = 2730. \)[/tex]
### 3. Find [tex]\( S_{16} \)[/tex] of the arithmetic sequence [tex]\( 5, 1, -3, -7, \ldots \)[/tex]
For this sequence:
- The first term ([tex]\( a \)[/tex]) is [tex]\( 5 \)[/tex].
- The common difference ([tex]\( d \)[/tex]) is [tex]\( 1 - 5 = -4 \)[/tex].
- The number of terms ([tex]\( n \)[/tex]) is [tex]\( 16 \)[/tex].
Using the formula:
[tex]\[ S_{16} = \frac{16}{2} \left(2 \cdot 5 + (16-1) \cdot (-4)\right) \][/tex]
[tex]\[ S_{16} = 8 \left(10 - 60\right) \][/tex]
[tex]\[ S_{16} = 8 \cdot (-50) \][/tex]
[tex]\[ S_{16} = -400 \][/tex]
Thus, [tex]\( S_{16} = -400. \)[/tex]
### 4. Find [tex]\( S_{36} \)[/tex] of the arithmetic sequence [tex]\( 12, 23, 34, 45, \ldots \)[/tex]
For this sequence:
- The first term ([tex]\( a \)[/tex]) is [tex]\( 12 \)[/tex].
- The common difference ([tex]\( d \)[/tex]) is [tex]\( 23 - 12 = 11 \)[/tex].
- The number of terms ([tex]\( n \)[/tex]) is [tex]\( 36 \)[/tex].
Using the formula:
[tex]\[ S_{36} = \frac{36}{2} \left(2 \cdot 12 + (36-1) \cdot 11\right) \][/tex]
[tex]\[ S_{36} = 18 \left(24 + 385\right) \][/tex]
[tex]\[ S_{36} = 18 \cdot 409 \][/tex]
[tex]\[ S_{36} = 7362 \][/tex]
Thus, [tex]\( S_{36} = 7362. \]
In conclusion, we have:
1. \( S_{20} = 1550 \)[/tex]
2. [tex]\( S_{30} = 2730 \)[/tex]
3. [tex]\( S_{16} = -400 \)[/tex]
4. [tex]\( S_{36} = 7362 \)[/tex]
