Certainly! Let's work through each sequence in detail, identify the missing numbers, and state the rule for each sequence:
1. Sequence: [tex]\( 16, 20, 24, 28, 32, 36, 40, 44 \)[/tex]
Rule: This sequence increases by 4 with each step. The rule can be expressed as:
[tex]\[
a_n = 16 + 4n
\][/tex]
This is an arithmetic sequence with a common difference of 4 starting at 16.
2. Sequence: [tex]\( 9, 5, 1, -3, -7, -11 \)[/tex]
Rule: This sequence decreases by 4 with each step. The rule can be expressed as:
[tex]\[
a_n = 9 - 4n
\][/tex]
This is an arithmetic sequence with a common difference of -4 starting at 9.
3. Sequence: [tex]\( 20, 30, 40, \_, 60, 70 \)[/tex]
Rule: This sequence increases by 10 with each step. The rule can be expressed as:
[tex]\[
a_n = 20 + 10n
\][/tex]
The missing number is 50. This is an arithmetic sequence with a common difference of 10 starting at 20.
4. Sequence: [tex]\( 15, -45, -135, \_, 1215, -3645 \)[/tex]
Rule: This sequence is geometric with each term being multiplied by -3. The rule can be expressed as:
[tex]\[
a_n = 15 \times (-3)^n
\][/tex]
The missing number is [tex]\( 405 \)[/tex]. This is a geometric sequence where the ratio between terms is -3.
5. Sequence: [tex]\( \frac{1}{3}, \frac{1}{8}, \frac{1}{22}, \frac{1}{26} \)[/tex]
Rule: This sequence appears to follow the pattern:
[tex]\[
a_n = \frac{1}{4n - 1}
\][/tex]
This is a sequence where the denominator increases in a specific arithmetic pattern.
So, filling in the missing details and rules, we have:
1. Sequence: [tex]\( 16, 20, 24, 28, 32, 36, 40, 44 \)[/tex]
Rule: [tex]\( a_n = 16 + 4n \)[/tex]
2. Sequence: [tex]\( 9, 5, 1, -3, -7, -11 \)[/tex]
Rule: [tex]\( a_n = 9 - 4n \)[/tex]
3. Sequence: [tex]\( 20, 30, 40, 50, 60, 70 \)[/tex]
Rule: [tex]\( a_n = 20 + 10n \)[/tex]
4. Sequence: [tex]\( 15, -45, -135, 405, -1215 \)[/tex]
Rule: [tex]\( a_n = 15 \times (-3)^n \)[/tex]
5. Sequence: [tex]\( \frac{1}{3}, \frac{1}{8}, \frac{1}{22}, \frac{1}{26} \)[/tex]
Rule: [tex]\( a_n = \frac{1}{4n - 1} \)[/tex]
