Alright, let's solve the problem step-by-step.
1. Finding the [tex]\( n \)[/tex]-th term of the arithmetic sequence [tex]\( 4, 10, 16, 22, \ldots \)[/tex]
An arithmetic sequence follows the formula of the [tex]\( n \)[/tex]-th term:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
Where:
- [tex]\( a_n \)[/tex] is the [tex]\( n \)[/tex]-th term,
- [tex]\( a_1 \)[/tex] is the first term,
- [tex]\( d \)[/tex] is the common difference,
- [tex]\( n \)[/tex] is the position of the term in the sequence.
Given:
- First term ([tex]\( a_1 \)[/tex]) = 4
- Common difference ([tex]\( d \)[/tex]) = 10 - 4 = 6
So, the [tex]\( n \)[/tex]-th term formula for this sequence is:
[tex]\[
a_n = 4 + (n - 1) \times 6 = 6n - 2
\][/tex]
2. Finding the [tex]\( n \)[/tex]-th term of the arithmetic sequence [tex]\( 2.3, -1, -4.3, \ldots \)[/tex]
Using the formula:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
Given:
- First term ([tex]\( a_1 \)[/tex]) = 2.3
- Common difference ([tex]\( d \)[/tex]) = -1 - 2.3 = -3.3
So, the [tex]\( n \)[/tex]-th term formula for this sequence is:
[tex]\[
a_n = 2.3 + (n - 1) \times (-3.3) = 2.3 - 3.3n + 3.3 - 1
\][/tex]
It simplifies to:
[tex]\[
a_n = 5.6 - 3.3n
\][/tex]
3. Finding the [tex]\( n \)[/tex]-th term of the arithmetic sequence [tex]\( -11, -9, -7, -5, \ldots \)[/tex]
Using the formula:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
Given:
- First term ([tex]\( a_1 \)[/tex]) = -11
- Common difference ([tex]\( d \)[/tex]) = -9 - (-11) = 2
So, the [tex]\( n \)[/tex]-th term formula for this sequence is:
[tex]\[
a_n = -11 + (n - 1) \times 2 = -11 + 2n - 2
\][/tex]
It simplifies to:
[tex]\[
a_n = 2n - 13
\][/tex]
4. Finding the 20th term of the arithmetic sequence [tex]\( 45, 8, 11, 14, \ldots \)[/tex]
For this sequence, we recognize the pattern only includes a sequence [tex]\( 45, 8, 11, 14 \)[/tex]. To find the common difference and terms:
Given:
- First term ([tex]\( a_1 \)[/tex]) = 45
- Common difference ([tex]\( d \)[/tex]) ≈ [tex]\( d \)[/tex] calculated from next element.
Corrected:
[tex]\[ 8 - 45 = d = -37 \][/tex]
So, the formula:
[tex]\[ a_5 \][/tex]
Given 45th term:
[tex]\[ a_2 as a compare 20th term :
\][/tex]
[tex]\[
a_{20} = 45 + (20 - 1) \times (-37)
= 45 - 37 \times 19 = -658
]
5. Finding the next term of the arithmetic sequence \( 20, 16, 12, 8, \ldots \)
Given:
- Common difference (\( d \)) = 16 - 20 = -4
Next term after \( 8 \):
\[
a_{next} = 8 - 4 = 4
\][/tex]
So, the next term in the sequence is \(
Next term: 4
]
