Question

question
find the 10th term of the geometric sequence 10, -20, 40, ...

Explanation:

Step1: Identify the first term and common ratio

The first term \(a_1\) of the geometric sequence is \(10\). To find the common ratio \(r\), divide the second term by the first term: \(r=\frac{-20}{10}=-2\).

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 \(n = 10\), \(a_1 = 10\), and \(r=-2\). Substitute these values into the formula: \(a_{10}=10\times(-2)^{10 - 1}\).

Step3: Calculate the exponent and the term

First, calculate the exponent: \(10 - 1 = 9\). Then, calculate \((-2)^9=-512\). Now, multiply by the first term: \(a_{10}=10\times(-512)=-5120\).

Answer:

\(-5120\)