Given the sequence 3, 6, 12, 24, 48, 96 …

Question

Explain what makes it geometric.

It is geometric because you are _ _ each time.

Write a recursive formula an = an-1r for the geometric sequence.

a1 + _
an = a(n-1)⋅.

Write an explicit formula (an = a1r^(n-1)) for the geometric sequence. Use your explicit formula to find the 32nd term of the sequence.

an = _⋅_^(n-1)
a32 = _⋅_^(_-1)
a32 = _

Answer 1

The given sequence is 3, 6, 12, 24, 48, 96 ...

What makes it geometric:

It is geometric because you are multiplying by 2 each time.

Recursive Formula:

The recursive formula for a geometric sequence is given by: \[ a_n = a_{n-1} \cdot r \]

For this sequence, we have:

• \( a_1 = 3 \) (the first term)
• The common ratio \( r = 2 \)

So the recursive formula can be written as: \[ a_n = a_{n-1} \cdot 2 \]

Explicit Formula:

The explicit formula for a geometric sequence is: \[ a_n = a_1 \cdot r^{(n-1)} \]

Using the initial term \( a_1 = 3 \) and common ratio \( r = 2 \), we have: \[ a_n = 3 \cdot 2^{(n-1)} \]

Finding the 32nd Term:

To find the 32nd term, we substitute \( n = 32 \) into the explicit formula: \[ a_{32} = 3 \cdot 2^{(32-1)} \] \[ a_{32} = 3 \cdot 2^{31} \]

Final Calculation of a32:

Calculating \( 2^{31} \):

1. \( 2^{31} = 2147483648 \)
2. Hence, \( a_{32} = 3 \cdot 2147483648 \)
3. Therefore, \( a_{32} = 6442450944 \)

Final Answers:

• Recursive formula: \( a_n = a_{n-1} \cdot 2 \)
• Explicit formula: \( a_n = 3 \cdot 2^{(n-1)} \)
• \( a_{32} = 6442450944 \)