Answer :
Let's solve each sequence step-by-step to determine the first five terms.
### 1. Sequence: \( a_{n+1} = a_n + 6 \), starting with \( a_1 = 11 \) for \( n \geq 1 \)
We start with the first term \( a_1 \):
[tex]\[ a_1 = 11 \][/tex]
Now, let's find the subsequent terms:
- For \( n = 1 \): \( a_2 = a_1 + 6 = 11 + 6 = 17 \)
- For \( n = 2 \): \( a_3 = a_2 + 6 = 17 + 6 = 23 \)
- For \( n = 3 \): \( a_4 = a_3 + 6 = 23 + 6 = 29 \)
- For \( n = 4 \): \( a_5 = a_4 + 6 = 29 + 6 = 35 \)
So, the first five terms of this sequence are:
[tex]\[ 11, 17, 23, 29, 35 \][/tex]
### 2. Sequence: \( a_n = a_{n-1} \div 2 \), starting with \( a_1 = 50 \) for \( n \geq 2 \)
We start with the first term \( a_1 \):
[tex]\[ a_1 = 50 \][/tex]
Now, let's find the subsequent terms:
- For \( n = 1 \): \( a_2 = a_1 \div 2 = 50 \div 2 = 25.0 \)
- For \( n = 2 \): \( a_3 = a_2 \div 2 = 25.0 \div 2 = 12.5 \)
- For \( n = 3 \): \( a_4 = a_3 \div 2 = 12.5 \div 2 = 6.25 \)
- For \( n = 4 \): \( a_5 = a_4 \div 2 = 6.25 \div 2 = 3.125 \)
So, the first five terms of this sequence are:
[tex]\[ 50, 25.0, 12.5, 6.25, 3.125 \][/tex]
### 3. Sequence: \( f(n+1) = -2 f(n) + 8 \), starting with \( f(1) = 1 \) for \( n \geq 1 \)
We start with the first term \( f(1) \):
[tex]\[ f(1) = 1 \][/tex]
Now, let's find the subsequent terms:
- For \( n = 1 \): \( f(2) = -2 f(1) + 8 = -2 \cdot 1 + 8 = 6 \)
- For \( n = 2 \): \( f(3) = -2 f(2) + 8 = -2 \cdot 6 + 8 = -12 + 8 = -4 \)
- For \( n = 3 \): \( f(4) = -2 f(3) + 8 = -2 \cdot (-4) + 8 = 8 + 8 = 16 \)
- For \( n = 4 \): \( f(5) = -2 f(4) + 8 = -2 \cdot 16 + 8 = -32 + 8 = -24 \)
So, the first five terms of this sequence are:
[tex]\[ 1, 6, -4, 16, -24 \][/tex]
### Summary
The first five terms for each sequence are:
1. Sequence \( a_{n+1} = a_n + 6 \):
[tex]\[ 11, 17, 23, 29, 35 \][/tex]
2. Sequence \( a_n = a_{n-1} \div 2 \):
[tex]\[ 50, 25.0, 12.5, 6.25, 3.125 \][/tex]
3. Sequence \( f(n+1) = -2 f(n) + 8 \):
[tex]\[ 1, 6, -4, 16, -24 \][/tex]
### 1. Sequence: \( a_{n+1} = a_n + 6 \), starting with \( a_1 = 11 \) for \( n \geq 1 \)
We start with the first term \( a_1 \):
[tex]\[ a_1 = 11 \][/tex]
Now, let's find the subsequent terms:
- For \( n = 1 \): \( a_2 = a_1 + 6 = 11 + 6 = 17 \)
- For \( n = 2 \): \( a_3 = a_2 + 6 = 17 + 6 = 23 \)
- For \( n = 3 \): \( a_4 = a_3 + 6 = 23 + 6 = 29 \)
- For \( n = 4 \): \( a_5 = a_4 + 6 = 29 + 6 = 35 \)
So, the first five terms of this sequence are:
[tex]\[ 11, 17, 23, 29, 35 \][/tex]
### 2. Sequence: \( a_n = a_{n-1} \div 2 \), starting with \( a_1 = 50 \) for \( n \geq 2 \)
We start with the first term \( a_1 \):
[tex]\[ a_1 = 50 \][/tex]
Now, let's find the subsequent terms:
- For \( n = 1 \): \( a_2 = a_1 \div 2 = 50 \div 2 = 25.0 \)
- For \( n = 2 \): \( a_3 = a_2 \div 2 = 25.0 \div 2 = 12.5 \)
- For \( n = 3 \): \( a_4 = a_3 \div 2 = 12.5 \div 2 = 6.25 \)
- For \( n = 4 \): \( a_5 = a_4 \div 2 = 6.25 \div 2 = 3.125 \)
So, the first five terms of this sequence are:
[tex]\[ 50, 25.0, 12.5, 6.25, 3.125 \][/tex]
### 3. Sequence: \( f(n+1) = -2 f(n) + 8 \), starting with \( f(1) = 1 \) for \( n \geq 1 \)
We start with the first term \( f(1) \):
[tex]\[ f(1) = 1 \][/tex]
Now, let's find the subsequent terms:
- For \( n = 1 \): \( f(2) = -2 f(1) + 8 = -2 \cdot 1 + 8 = 6 \)
- For \( n = 2 \): \( f(3) = -2 f(2) + 8 = -2 \cdot 6 + 8 = -12 + 8 = -4 \)
- For \( n = 3 \): \( f(4) = -2 f(3) + 8 = -2 \cdot (-4) + 8 = 8 + 8 = 16 \)
- For \( n = 4 \): \( f(5) = -2 f(4) + 8 = -2 \cdot 16 + 8 = -32 + 8 = -24 \)
So, the first five terms of this sequence are:
[tex]\[ 1, 6, -4, 16, -24 \][/tex]
### Summary
The first five terms for each sequence are:
1. Sequence \( a_{n+1} = a_n + 6 \):
[tex]\[ 11, 17, 23, 29, 35 \][/tex]
2. Sequence \( a_n = a_{n-1} \div 2 \):
[tex]\[ 50, 25.0, 12.5, 6.25, 3.125 \][/tex]
3. Sequence \( f(n+1) = -2 f(n) + 8 \):
[tex]\[ 1, 6, -4, 16, -24 \][/tex]
