Let's break down the steps to find the first five terms of each of the given sequences:
Sequence (a): [tex]\( T(n+1) = T(n) + 4 \)[/tex] with [tex]\( T(1) = 5 \)[/tex]
1. [tex]\( T(1) = 5 \)[/tex]
2. [tex]\( T(2) = T(1) + 4 = 5 + 4 = 9 \)[/tex]
3. [tex]\( T(3) = T(2) + 4 = 9 + 4 = 13 \)[/tex]
4. [tex]\( T(4) = T(3) + 4 = 13 + 4 = 17 \)[/tex]
5. [tex]\( T(5) = T(4) + 4 = 17 + 4 = 21 \)[/tex]
So, the first five terms are: [tex]\([5, 9, 13, 17, 21]\)[/tex]
Sequence (b): [tex]\( T(n+1) = T(n) + 6 \)[/tex] with [tex]\( T(1) = 0 \)[/tex]
1. [tex]\( T(1) = 0 \)[/tex]
2. [tex]\( T(2) = T(1) + 6 = 0 + 6 = 6 \)[/tex]
3. [tex]\( T(3) = T(2) + 6 = 6 + 6 = 12 \)[/tex]
4. [tex]\( T(4) = T(3) + 6 = 12 + 6 = 18 \)[/tex]
5. [tex]\( T(5) = T(4) + 6 = 18 + 6 = 24 \)[/tex]
So, the first five terms are: [tex]\([0, 6, 12, 18, 24]\)[/tex]
Sequence (c): [tex]\( T(n+1) = T(n) - 3 \)[/tex] with [tex]\( T(1) = 5 \)[/tex]
1. [tex]\( T(1) = 5 \)[/tex]
2. [tex]\( T(2) = T(1) - 3 = 5 - 3 = 2 \)[/tex]
3. [tex]\( T(3) = T(2) - 3 = 2 - 3 = -1 \)[/tex]
4. [tex]\( T(4) = T(3) - 3 = -1 - 3 = -4 \)[/tex]
5. [tex]\( T(5) = T(4) - 3 = -4 - 3 = -7 \)[/tex]
So, the first five terms are: [tex]\([5, 2, -1, -4, -7]\)[/tex]
Sequence (d): [tex]\( T(n+1) = T(n) - 2 \)[/tex] with [tex]\( T(1) = -3 \)[/tex]
1. [tex]\( T(1) = -3 \)[/tex]
2. [tex]\( T(2) = T(1) - 2 = -3 - 2 = -5 \)[/tex]
3. [tex]\( T(3) = T(2) - 2 = -5 - 2 = -7 \)[/tex]
4. [tex]\( T(4) = T(3) - 2 = -7 - 2 = -9 \)[/tex]
5. [tex]\( T(5) = T(4) - 2 = -9 - 2 = -11 \)[/tex]
So, the first five terms are: [tex]\([-3, -5, -7, -9, -11]\)[/tex]
Sequence (e): [tex]\( T(n+1) = T(n) - 6 \)[/tex] with [tex]\( T(1) = 1 \)[/tex]
1. [tex]\( T(1) = 1 \)[/tex]
2. [tex]\( T(2) = T(1) - 6 = 1 - 6 = -5 \)[/tex]
3. [tex]\( T(3) = T(2) - 6 = -5 - 6 = -11 \)[/tex]
4. [tex]\( T(4) = T(3) - 6 = -11 - 6 = -17 \)[/tex]
5. [tex]\( T(5) = T(4) - 6 = -17 - 6 = -23 \)[/tex]
So, the first five terms are: [tex]\([1, -5, -11, -17, -23]\)[/tex]
Sequence (f): [tex]\( T(n+1) = 10 - T(n) \)[/tex] with [tex]\( T(1) = 2 \)[/tex]
1. [tex]\( T(1) = 2 \)[/tex]
2. [tex]\( T(2) = 10 - T(1) = 10 - 2 = 8 \)[/tex]
3. [tex]\( T(3) = 10 - T(2) = 10 - 8 = 2 \)[/tex]
4. [tex]\( T(4) = 10 - T(3) = 10 - 2 = 8 \)[/tex]
5. [tex]\( T(5) = 10 - T(4) = 10 - 8 = 2 \)[/tex]
So, the first five terms are: [tex]\([2, 8, 2, 8, 2]\)[/tex]
In conclusion, the first five terms of each sequence are:
- a: [5, 9, 13, 17, 21]
- b: [0, 6, 12, 18, 24]
- c: [5, 2, -1, -4, -7]
- d: [-3, -5, -7, -9, -11]
- e: [1, -5, -11, -17, -23]
- f: [2, 8, 2, 8, 2]
