Write an equation for the $n$n th term of the geometric sequence. Then find $a_9$a9 .$5,-10,\ 20,-40,\ \ldots$5,−10, 20,−40, …$a_n=$an= $a_9=$a9=
Question
Write an equation for the th term of the geometric sequence. Then find .
The geometric sequence is:
Solution
The general formula for the nth term of a geometric sequence is:
a_n = a_1 * r^(n-1)
where:
- a_n is the nth term
- a_1 is the first term
- r is the common ratio
- n is the term number
In this sequence, the first term a_1 is 5 and the common ratio r is -2 (since each term is -2 times the previous term).
So, the nth term of the sequence can be written as:
a_n = 5 * (-2)^(n-1)
To find the 9th term a_9, we substitute n = 9 into the formula:
a_9 = 5 * (-2)^(9-1) a_9 = 5 * (-2)^8 a_9 = 5 * 256 a_9 = 1280
So, the 9th term of the sequence is 1280.
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