Question

find the 14th term of the geometric sequence 10, 20, 40, ...

Explanation:

Step1: Recall geometric sequence formula

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: Identify \(a_1\) and \(r\)

For the sequence \(10, 20, 40, \dots\), \(a_1 = 10\). The common ratio \(r=\frac{20}{10} = 2\).

Step3: Calculate the 14th term

Substitute \(a_1 = 10\), \(r = 2\), and \(n = 14\) into the formula: \(a_{14}=10\times2^{14 - 1}=10\times2^{13}\).
Calculate \(2^{13}=8192\), so \(a_{14}=10\times8192 = 81920\).

Answer:

\(81920\)