Question

nth term of a geometric sequence unique id: 3226
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question
find the 12th term of the geometric sequence 10, -50, 250, ...

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 \(r\), we divide the second term by the first term: \(r=\frac{-50}{10}=-5\). Also, we can check with the third term: \(\frac{250}{-50}=-5\), so \(r = - 5\) is consistent.

Step2: Use the formula for the nth term of a geometric sequence

The formula for the \(n\)-th term of a geometric sequence is \(a_n=a_1\times r^{n - 1}\). We want to find the 12th term, so \(n = 12\), \(a_1=10\), and \(r=-5\). Substitute these values into the formula: \(a_{12}=10\times(-5)^{12 - 1}\).

Step3: Calculate the exponent and then the term

First, calculate the exponent: \(12-1 = 11\). So we have \(a_{12}=10\times(-5)^{11}\). Now, \((-5)^{11}=-5^{11}\) (because the exponent is odd). \(5^{11}=48828125\), so \((-5)^{11}=-48828125\). Then, \(a_{12}=10\times(-48828125)=-488281250\).

Answer:

\(-488281250\)