Answer :
Sure! Let's go through each part of the question.
### Part 1: Write the First Five Terms of Each Sequence
1) Sequence: [tex]\( a_n = 3^{n-1} + 2 \)[/tex]
- For [tex]\( n=1 \)[/tex]: [tex]\( a_1 = 3^{1-1} + 2 = 3^0 + 2 = 1 + 2 = 3 \)[/tex]
- For [tex]\( n=2 \)[/tex]: [tex]\( a_2 = 3^{2-1} + 2 = 3^1 + 2 = 3 + 2 = 5 \)[/tex]
- For [tex]\( n=3 \)[/tex]: [tex]\( a_3 = 3^{3-1} + 2 = 3^2 + 2 = 9 + 2 = 11 \)[/tex]
- For [tex]\( n=4 \)[/tex]: [tex]\( a_4 = 3^{4-1} + 2 = 3^3 + 2 = 27 + 2 = 29 \)[/tex]
- For [tex]\( n=5 \)[/tex]: [tex]\( a_5 = 3^{5-1} + 2 = 3^4 + 2 = 81 + 2 = 83 \)[/tex]
- First five terms: 3, 5, 11, 29, 83
2) Sequence: [tex]\( a_n = -6n^3 \)[/tex]
- For [tex]\( n=1 \)[/tex]: [tex]\( a_1 = -6(1)^3 = -6 \)[/tex]
- For [tex]\( n=2 \)[/tex]: [tex]\( a_2 = -6(2)^3 = -48 \)[/tex]
- For [tex]\( n=3 \)[/tex]: [tex]\( a_3 = -6(3)^3 = -162 \)[/tex]
- For [tex]\( n=4 \)[/tex]: [tex]\( a_4 = -6(4)^3 = -384 \)[/tex]
- For [tex]\( n=5 \)[/tex]: [tex]\( a_5 = -6(5)^3 = -750 \)[/tex]
- First five terms: -6, -48, -162, -384, -750
3) Sequence: [tex]\( a_n = -4n^2 - 1 \)[/tex]
- For [tex]\( n=1 \)[/tex]: [tex]\( a_1 = -4(1)^2 - 1 = -5 \)[/tex]
- For [tex]\( n=2 \)[/tex]: [tex]\( a_2 = -4(2)^2 - 1 = -17 \)[/tex]
- For [tex]\( n=3 \)[/tex]: [tex]\( a_3 = -4(3)^2 - 1 = -37 \)[/tex]
- For [tex]\( n=4 \)[/tex]: [tex]\( a_4 = -4(4)^2 - 1 = -65 \)[/tex]
- For [tex]\( n=5 \)[/tex]: [tex]\( a_5 = -4(5)^2 - 1 = -101 \)[/tex]
- First five terms: -5, -17, -37, -65, -101
4) Sequence: [tex]\( a_n = \frac{7}{4n + 2} \)[/tex]
- For [tex]\( n=1 \)[/tex]: [tex]\( a_1 = \frac{7}{4(1) + 2} = \frac{7}{6} \approx 1.167 \)[/tex]
- For [tex]\( n=2 \)[/tex]: [tex]\( a_2 = \frac{7}{4(2) + 2} = \frac{7}{10} = 0.7 \)[/tex]
- For [tex]\( n=3 \)[/tex]: [tex]\( a_3 = \frac{7}{4(3) + 2} = \frac{7}{14} = 0.5 \)[/tex]
- For [tex]\( n=4 \)[/tex]: [tex]\( a_4 = \frac{7}{4(4) + 2} = \frac{7}{18} \approx 0.389 \)[/tex]
- For [tex]\( n=5 \)[/tex]: [tex]\( a_5 = \frac{7}{4(5) + 2} = \frac{7}{22} \approx 0.318 \)[/tex]
- First five terms: 1.167, 0.7, 0.5, 0.389, 0.318
### Part 2: Determine the Type of Sequences and Find Common Difference or Ratio
5) Sequence: 2, 6, 18, 54, ...
- This is a geometric sequence.
- Common ratio [tex]\( r \)[/tex]: [tex]\( \frac{6}{2} = 3 \)[/tex] (and similarly for other terms).
- The sequence is geometric with a common ratio of 3.
6) Sequence: 48, -12, 3, -[tex]\( \frac{3}{4} \)[/tex], ...
- This is a geometric sequence.
- Common ratio [tex]\( r \)[/tex]: [tex]\( \frac{-12}{48} = -\frac{1}{4} \)[/tex] (and similarly for other terms).
- The sequence is geometric with a common ratio of -0.25.
7) Sequence: [tex]\( \frac{11}{4}, \frac{12}{5}, \frac{13}{6}, \frac{14}{7}, ... \)[/tex]
- This sequence is neither arithmetic nor geometric since neither common difference nor common ratio is consistent.
- The sequence is neither arithmetic nor geometric.
8) Sequence: 103, 102.1, 101.2, 100.3, ...
- This is an arithmetic sequence.
- Common difference [tex]\( d \)[/tex]: [tex]\( 102.1 - 103 = -0.9 \)[/tex] (and similarly for other terms).
- The sequence is arithmetic with a common difference of -0.9.
I hope this explanation helps you understand how to analyze and solve these sequence problems! If you have any more questions, feel free to ask.
### Part 1: Write the First Five Terms of Each Sequence
1) Sequence: [tex]\( a_n = 3^{n-1} + 2 \)[/tex]
- For [tex]\( n=1 \)[/tex]: [tex]\( a_1 = 3^{1-1} + 2 = 3^0 + 2 = 1 + 2 = 3 \)[/tex]
- For [tex]\( n=2 \)[/tex]: [tex]\( a_2 = 3^{2-1} + 2 = 3^1 + 2 = 3 + 2 = 5 \)[/tex]
- For [tex]\( n=3 \)[/tex]: [tex]\( a_3 = 3^{3-1} + 2 = 3^2 + 2 = 9 + 2 = 11 \)[/tex]
- For [tex]\( n=4 \)[/tex]: [tex]\( a_4 = 3^{4-1} + 2 = 3^3 + 2 = 27 + 2 = 29 \)[/tex]
- For [tex]\( n=5 \)[/tex]: [tex]\( a_5 = 3^{5-1} + 2 = 3^4 + 2 = 81 + 2 = 83 \)[/tex]
- First five terms: 3, 5, 11, 29, 83
2) Sequence: [tex]\( a_n = -6n^3 \)[/tex]
- For [tex]\( n=1 \)[/tex]: [tex]\( a_1 = -6(1)^3 = -6 \)[/tex]
- For [tex]\( n=2 \)[/tex]: [tex]\( a_2 = -6(2)^3 = -48 \)[/tex]
- For [tex]\( n=3 \)[/tex]: [tex]\( a_3 = -6(3)^3 = -162 \)[/tex]
- For [tex]\( n=4 \)[/tex]: [tex]\( a_4 = -6(4)^3 = -384 \)[/tex]
- For [tex]\( n=5 \)[/tex]: [tex]\( a_5 = -6(5)^3 = -750 \)[/tex]
- First five terms: -6, -48, -162, -384, -750
3) Sequence: [tex]\( a_n = -4n^2 - 1 \)[/tex]
- For [tex]\( n=1 \)[/tex]: [tex]\( a_1 = -4(1)^2 - 1 = -5 \)[/tex]
- For [tex]\( n=2 \)[/tex]: [tex]\( a_2 = -4(2)^2 - 1 = -17 \)[/tex]
- For [tex]\( n=3 \)[/tex]: [tex]\( a_3 = -4(3)^2 - 1 = -37 \)[/tex]
- For [tex]\( n=4 \)[/tex]: [tex]\( a_4 = -4(4)^2 - 1 = -65 \)[/tex]
- For [tex]\( n=5 \)[/tex]: [tex]\( a_5 = -4(5)^2 - 1 = -101 \)[/tex]
- First five terms: -5, -17, -37, -65, -101
4) Sequence: [tex]\( a_n = \frac{7}{4n + 2} \)[/tex]
- For [tex]\( n=1 \)[/tex]: [tex]\( a_1 = \frac{7}{4(1) + 2} = \frac{7}{6} \approx 1.167 \)[/tex]
- For [tex]\( n=2 \)[/tex]: [tex]\( a_2 = \frac{7}{4(2) + 2} = \frac{7}{10} = 0.7 \)[/tex]
- For [tex]\( n=3 \)[/tex]: [tex]\( a_3 = \frac{7}{4(3) + 2} = \frac{7}{14} = 0.5 \)[/tex]
- For [tex]\( n=4 \)[/tex]: [tex]\( a_4 = \frac{7}{4(4) + 2} = \frac{7}{18} \approx 0.389 \)[/tex]
- For [tex]\( n=5 \)[/tex]: [tex]\( a_5 = \frac{7}{4(5) + 2} = \frac{7}{22} \approx 0.318 \)[/tex]
- First five terms: 1.167, 0.7, 0.5, 0.389, 0.318
### Part 2: Determine the Type of Sequences and Find Common Difference or Ratio
5) Sequence: 2, 6, 18, 54, ...
- This is a geometric sequence.
- Common ratio [tex]\( r \)[/tex]: [tex]\( \frac{6}{2} = 3 \)[/tex] (and similarly for other terms).
- The sequence is geometric with a common ratio of 3.
6) Sequence: 48, -12, 3, -[tex]\( \frac{3}{4} \)[/tex], ...
- This is a geometric sequence.
- Common ratio [tex]\( r \)[/tex]: [tex]\( \frac{-12}{48} = -\frac{1}{4} \)[/tex] (and similarly for other terms).
- The sequence is geometric with a common ratio of -0.25.
7) Sequence: [tex]\( \frac{11}{4}, \frac{12}{5}, \frac{13}{6}, \frac{14}{7}, ... \)[/tex]
- This sequence is neither arithmetic nor geometric since neither common difference nor common ratio is consistent.
- The sequence is neither arithmetic nor geometric.
8) Sequence: 103, 102.1, 101.2, 100.3, ...
- This is an arithmetic sequence.
- Common difference [tex]\( d \)[/tex]: [tex]\( 102.1 - 103 = -0.9 \)[/tex] (and similarly for other terms).
- The sequence is arithmetic with a common difference of -0.9.
I hope this explanation helps you understand how to analyze and solve these sequence problems! If you have any more questions, feel free to ask.
