Question

question
find the 10th term of the geometric sequence 3, 15, 75, ...

Explanation:

Step1: Identify the first term and common ratio

In a geometric sequence, the first term \(a_1 = 3\). The common ratio \(r\) is found by dividing the second term by the first term: \(r=\frac{15}{3} = 5\).

Step2: Use the geometric sequence formula

The formula for the \(n\)-th term of a geometric sequence is \(a_n=a_1\times r^{n - 1}\). We need to find the 10th term, so \(n = 10\), \(a_1=3\), and \(r = 5\).
Substitute these values into the formula: \(a_{10}=3\times5^{10 - 1}\).

Step3: Simplify the expression

First, calculate the exponent: \(10-1 = 9\). Then, calculate \(5^9\). \(5^9=5\times5\times5\times5\times5\times5\times5\times5\times5 = 1953125\).
Now, multiply by 3: \(a_{10}=3\times1953125 = 5859375\).

Answer:

\(5859375\)