What is the 17th term of the geometric progression: 2, 2√2, 4, 4√2, ...?
Understand the Problem
The question is asking to find the 17th term of the given geometric progression (G.P.). The G.P. is 2, 2√2, 4, 4√2,... We need to determine the common ratio and then use the formula for the nth term of a G.P. to find the 17th term.
Answer
$512$
Answer for screen readers
$512$
Steps to Solve
- Find the common ratio
To find the common ratio ($r$) of the geometric progression, divide any term by its preceding term.
$r = \frac{2\sqrt{2}}{2} = \sqrt{2}$
Alternatively: $r = \frac{4}{2\sqrt{2}} = \frac{2}{\sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2}$
- Find the 17th term ($a_{17}$)
The formula for the $n$th term of a G.P. is $a_n = a_1 \cdot r^{(n-1)}$, where $a_1$ is the first term and $r$ is the common ratio. In this case, $a_1 = 2$, $r = \sqrt{2}$, and $n = 17$.
$a_{17} = 2 \cdot (\sqrt{2})^{(17-1)}$ $a_{17} = 2 \cdot (\sqrt{2})^{16}$ $a_{17} = 2 \cdot (2^{1/2})^{16}$ $a_{17} = 2 \cdot 2^{8}$ $a_{17} = 2 \cdot 256$ $a_{17} = 512$
$512$
More Information
The 17th term of the given geometric progression is 512.
Tips
- Forgetting the formula for the $n$th term of a geometric progression. Remember $a_n = a_1 \cdot r^{n-1}$
- Incorrectly calculating the common ratio. Dividing a term by its preceding term yields the common ratio.
- Making mistakes in simplifying the powers. e.g. $(\sqrt{2})^{16} = (2^{1/2})^{16} = 2^8$
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