Answered

Write the first five terms of each sequence. Determine whether each sequence is arithmetic, geometric, or neither.

[tex]\[
\begin{array}{l}
a(1)=7, \ a(n)=a(n-1)-3 \text{ for } n \geq 2. \\
b(1)=2, \ b(n)=2 \cdot b(n-1)-1 \text{ for } n \geq 2. \\
c(1)=3, \ c(n)=10 \cdot c(n-1) \text{ for } n \geq 2. \\
d(1)=1, \ d(n)=n \cdot d(n-1) \text{ for } n \geq 2.
\end{array}
\][/tex]



Answer :

GinnyAnswer
Let's examine each sequence one by one, find its first five terms, and determine its type.

### Sequence [tex]\( a(n) \)[/tex]
- Initial term: [tex]\( a(1) = 7 \)[/tex]
- Recurrence relation: [tex]\( a(n) = a(n-1) - 3 \)[/tex] for [tex]\( n \geq 2 \)[/tex]

Let's calculate the first five terms:
1. [tex]\( a(1) = 7 \)[/tex]
2. [tex]\( a(2) = a(1) - 3 = 7 - 3 = 4 \)[/tex]
3. [tex]\( a(3) = a(2) - 3 = 4 - 3 = 1 \)[/tex]
4. [tex]\( a(4) = a(3) - 3 = 1 - 3 = -2 \)[/tex]
5. [tex]\( a(5) = a(4) - 3 = -2 - 3 = -5 \)[/tex]

The first five terms are: [tex]\([7, 4, 1, -2, -5]\)[/tex].

To determine the type, observe the differences between consecutive terms:
[tex]\[ 4 - 7 = -3 \][/tex]
[tex]\[ 1 - 4 = -3 \][/tex]
[tex]\[ -2 - 1 = -3 \][/tex]
[tex]\[ -5 - (-2) = -3 \][/tex]

Since the differences are constant, this sequence is arithmetic.

### Sequence [tex]\( b(n) \)[/tex]
- Initial term: [tex]\( b(1) = 2 \)[/tex]
- Recurrence relation: [tex]\( b(n) = 2 \cdot b(n-1) - 1 \)[/tex] for [tex]\( n \geq 2 \)[/tex]

Let's calculate the first five terms:
1. [tex]\( b(1) = 2 \)[/tex]
2. [tex]\( b(2) = 2 \cdot b(1) - 1 = 2 \cdot 2 - 1 = 3 \)[/tex]
3. [tex]\( b(3) = 2 \cdot b(2) - 1 = 2 \cdot 3 - 1 = 5 \)[/tex]
4. [tex]\( b(4) = 2 \cdot b(3) - 1 = 2 \cdot 5 - 1 = 9 \)[/tex]
5. [tex]\( b(5) = 2 \cdot b(4) - 1 = 2 \cdot 9 - 1 = 17 \)[/tex]

The first five terms are: [tex]\([2, 3, 5, 9, 17]\)[/tex].

This sequence does not have constant differences (arithmetic) nor constant ratios (geometric), so it is neither.

### Sequence [tex]\( c(n) \)[/tex]
- Initial term: [tex]\( c(1) = 3 \)[/tex]
- Recurrence relation: [tex]\( c(n) = 10 \cdot c(n-1) \)[/tex] for [tex]\( n \geq 2 \)[/tex]

Let's calculate the first five terms:
1. [tex]\( c(1) = 3 \)[/tex]
2. [tex]\( c(2) = 10 \cdot c(1) = 10 \cdot 3 = 30 \)[/tex]
3. [tex]\( c(3) = 10 \cdot c(2) = 10 \cdot 30 = 300 \)[/tex]
4. [tex]\( c(4) = 10 \cdot c(3) = 10 \cdot 300 = 3000 \)[/tex]
5. [tex]\( c(5) = 10 \cdot c(4) = 10 \cdot 3000 = 30000 \)[/tex]

The first five terms are: [tex]\([3, 30, 300, 3000, 30000]\)[/tex].

To determine the type, observe the ratios between consecutive terms:
[tex]\[ \frac{30}{3} = 10 \][/tex]
[tex]\[ \frac{300}{30} = 10 \][/tex]
[tex]\[ \frac{3000}{300} = 10 \][/tex]
[tex]\[ \frac{30000}{3000} = 10 \][/tex]

Since the ratios are constant, this sequence is geometric.

### Sequence [tex]\( d(n) \)[/tex]
- Initial term: [tex]\( d(1) = 1 \)[/tex]
- Recurrence relation: [tex]\( d(n) = n \cdot d(n-1) \)[/tex] for [tex]\( n \geq 2 \)[/tex]

Let's calculate the first five terms:
1. [tex]\( d(1) = 1 \)[/tex]
2. [tex]\( d(2) = 2 \cdot d(1) = 2 \cdot 1 = 2 \)[/tex]
3. [tex]\( d(3) = 3 \cdot d(2) = 3 \cdot 2 = 6 \)[/tex]
4. [tex]\( d(4) = 4 \cdot d(3) = 4 \cdot 6 = 24 \)[/tex]
5. [tex]\( d(5) = 5 \cdot d(4) = 5 \cdot 24 = 120 \)[/tex]

The first five terms are: [tex]\([1, 2, 6, 24, 120]\)[/tex].

This sequence does not have constant differences (arithmetic) nor constant ratios (geometric), so it is neither.

### Summary
- Sequence [tex]\( a(n) \)[/tex]: First five terms: [tex]\([7, 4, 1, -2, -5]\)[/tex], Type: arithmetic
- Sequence [tex]\( b(n) \)[/tex]: First five terms: [tex]\([2, 3, 5, 9, 17]\)[/tex], Type: neither
- Sequence [tex]\( c(n) \)[/tex]: First five terms: [tex]\([3, 30, 300, 3000, 30000]\)[/tex], Type: geometric
- Sequence [tex]\( d(n) \)[/tex]: First five terms: [tex]\([1, 2, 6, 24, 120]\)[/tex], Type: neither

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