Question

exponents and exponential functions
finding a specified term of a geometric sequence give...
find the 11^th term of the following geometric sequence.
3, 15, 75, 375, ...

Explanation:

Step1: Identify the formula for a geometric sequence

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

Step2: Find the common ratio \(r\)

To find \(r\), divide the second term by the first term: \(r=\frac{15}{3} = 5\).

Step3: Identify \(a_1\) and \(n\)

Here, \(a_1 = 3\) and \(n = 11\) (since we need the 11th term).

Step4: Substitute into the formula

Substitute \(a_1 = 3\), \(r = 5\), and \(n = 11\) into the formula: \(a_{11}=3\cdot5^{11 - 1}\).

Step5: Simplify the exponent

Simplify \(11 - 1 = 10\), so \(a_{11}=3\cdot5^{10}\).

Step6: Calculate \(5^{10}\)

\(5^{10}=9765625\).

Step7: Multiply by 3

\(a_{11}=3\times9765625 = 29296875\).

Answer:

\(29296875\)