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Determine the recursive formula and the general term for the geometric sequence in which:

a) The first term is 19 and the common ratio is 5.
b) [tex]\( t_1 = -9 \)[/tex] and [tex]\( r = -4 \)[/tex].
c) The first term is 144 and the second term is 36.
d) [tex]\( t_1 = 900 \)[/tex] and [tex]\( r = \frac{1}{6} \)[/tex].

Sagot :

GinnyAnswer
To solve these problems, we need to determine both the recursive formula and the general term for each geometric sequence. We'll approach each part separately.

### Part (a)

#### Given:
- The first term [tex]\( a_1 = 19 \)[/tex]
- The common ratio [tex]\( r = 5 \)[/tex]

#### Recursive Formula:
The recursive formula for a geometric sequence is given by:
[tex]\[ a_n = a_{n-1} \times r \][/tex]
So, for this sequence:
[tex]\[ a_n = a_{n-1} \times 5 \][/tex]

#### General Term:
The general term for a geometric sequence is:
[tex]\[ a_n = a_1 \times r^{n-1} \][/tex]
Substituting the given values:
[tex]\[ a_n = 19 \times 5^{n-1} \][/tex]

### Part (b)

#### Given:
- The first term [tex]\( t_1 = -9 \)[/tex]
- The common ratio [tex]\( r = -4 \)[/tex]

#### Recursive Formula:
The recursive formula is:
[tex]\[ a_n = a_{n-1} \times (-4) \][/tex]

#### General Term:
The general term is:
[tex]\[ a_n = -9 \times (-4)^{n-1} \][/tex]

### Part (c)

#### Given:
- The first term [tex]\( a_1 = 144 \)[/tex]
- The second term [tex]\( a_2 = 36 \)[/tex]

To find the common ratio [tex]\( r \)[/tex]:
[tex]\[ r = \frac{a_2}{a_1} = \frac{36}{144} = \frac{1}{4} \][/tex]

#### Recursive Formula:
The recursive formula is:
[tex]\[ a_n = a_{n-1} \times \frac{1}{4} \][/tex]

#### General Term:
The general term is:
[tex]\[ a_n = 144 \times \left( \frac{1}{4} \right)^{n-1} \][/tex]

### Part (d)

#### Given:
- The first term [tex]\( t_1 = 900 \)[/tex]
- The common ratio [tex]\( r = \frac{1}{6} \)[/tex]

#### Recursive Formula:
The recursive formula is:
[tex]\[ a_n = a_{n-1} \times \frac{1}{6} \][/tex]

#### General Term:
The general term is:
[tex]\[ a_n = 900 \times \left( \frac{1}{6} \right)^{n-1} \][/tex]

### Summary of Results

(a) For the sequence with first term 19 and common ratio 5:
- Recursive formula: [tex]\( a_n = a_{n-1} \times 5 \)[/tex]
- General term: [tex]\( a_n = 19 \times 5^{n-1} \)[/tex]

(b) For the sequence with [tex]\( t_1 = -9 \)[/tex] and [tex]\( r = -4 \)[/tex]:
- Recursive formula: [tex]\( a_n = a_{n-1} \times (-4) \)[/tex]
- General term: [tex]\( a_n = -9 \times (-4)^{n-1} \)[/tex]

(c) For the sequence with first term 144 and second term 36:
- Recursive formula: [tex]\( a_n = a_{n-1} \times \frac{1}{4} \)[/tex]
- General term: [tex]\( a_n = 144 \times \left( \frac{1}{4} \right)^{n-1} \)[/tex]

(d) For the sequence with [tex]\( t_1 = 900 \)[/tex] and [tex]\( r = \frac{1}{6} \)[/tex]:
- Recursive formula: [tex]\( a_n = a_{n-1} \times \frac{1}{6} \)[/tex]
- General term: [tex]\( a_n = 900 \times \left( \frac{1}{6} \right)^{n-1} \)[/tex]
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