Question

write a formula for the nth term of the following ge
3, 12, 48, ...

enter a formula for the nth term of the geometric sequ
(a_n = 3 cdot (4)^{n - 1})

enter the 6th term of the geometric sequence.
(a_6 = square) (simplify your answer.)

Explanation:

Step1: Recall the formula for the nth term of a geometric sequence

The formula for the nth term of a geometric sequence is \(a_n = a_1 \cdot r^{n - 1}\), where \(a_1\) is the first term and \(r\) is the common ratio. Here, \(a_1 = 3\) and \(r = 4\), so the formula is \(a_n = 3 \cdot 4^{n - 1}\).

Step2: Substitute \(n = 6\) into the formula

To find the 6th term, we substitute \(n = 6\) into the formula \(a_n = 3 \cdot 4^{n - 1}\). So we have \(a_6 = 3 \cdot 4^{6 - 1}\).

Step3: Simplify the exponent

First, simplify the exponent: \(6 - 1 = 5\). So now we have \(a_6 = 3 \cdot 4^5\).

Step4: Calculate \(4^5\)

Calculate \(4^5\): \(4^5 = 4 \times 4 \times 4 \times 4 \times 4 = 1024\).

Step5: Multiply by 3

Now multiply 3 by 1024: \(3 \times 1024 = 3072\).

Answer:

\(a_6 = 3072\)