Question

find the 6th term of this geometric sequence.

(3, -12, 48, \dots)

(a_6 = ?)

hint: (a_n = a_1 \cdot r^{(n-1)})

Explanation:

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<reasoning>

Identify the first term and target term

We are given the geometric sequence:
\[3, -12, 48, \dots\]
The first term of the sequence is:
\[a_1 = 3\]
We want to find the 6th term, which means:
\[n = 6\]

Calculate the common ratio

To find the common ratio \(r\), we divide any term by its preceding term:
\[r = \frac{a_2}{a_1} = \frac{-12}{3} = -4\]
We can verify this with the next term:
\[\frac{a_3}{a_2} = \frac{48}{-12} = -4\]
Thus, the common ratio is \(r = -4\).

Apply the geometric sequence formula

The formula for the \(n\)-th term of a geometric sequence is:
\[a_n = a_1 \cdot r^{(n-1)}\]
Substituting our known values \(a_1 = 3\), \(r = -4\), and \(n = 6\):
\[a_6 = 3 \cdot (-4)^{(6-1)}\]
\[a_6 = 3 \cdot (-4)^5\]

Compute the final value

First, calculate the exponent:
\[(-4)^5 = -1024\]
Now, multiply by the first term:
\[a_6 = 3 \cdot (-1024) = -3072\]
</reasoning>

<answer>
Find the 6th term of this geometric sequence.

\(3, -12, 48, \dots\)

\(a_6 =\) <blank>-3072</blank>
</answer>

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Answer:

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<reasoning>

Identify the first term and target term

We are given the geometric sequence:
\[3, -12, 48, \dots\]
The first term of the sequence is:
\[a_1 = 3\]
We want to find the 6th term, which means:
\[n = 6\]

Calculate the common ratio

To find the common ratio \(r\), we divide any term by its preceding term:
\[r = \frac{a_2}{a_1} = \frac{-12}{3} = -4\]
We can verify this with the next term:
\[\frac{a_3}{a_2} = \frac{48}{-12} = -4\]
Thus, the common ratio is \(r = -4\).

Apply the geometric sequence formula

The formula for the \(n\)-th term of a geometric sequence is:
\[a_n = a_1 \cdot r^{(n-1)}\]
Substituting our known values \(a_1 = 3\), \(r = -4\), and \(n = 6\):
\[a_6 = 3 \cdot (-4)^{(6-1)}\]
\[a_6 = 3 \cdot (-4)^5\]

Compute the final value

First, calculate the exponent:
\[(-4)^5 = -1024\]
Now, multiply by the first term:
\[a_6 = 3 \cdot (-1024) = -3072\]
</reasoning>

<answer>
Find the 6th term of this geometric sequence.

\(3, -12, 48, \dots\)

\(a_6 =\) <blank>-3072</blank>
</answer>

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