Question

question
find the 14th term of the geometric sequence 5, -10, 20, ...
answer attempt 1 out of 3

Explanation:

Step1: Identify $a_1$ and $r$

For a geometric sequence, the first term $a_1 = 5$. The common ratio $r$ is found by dividing the second term by the first term: $r=\frac{- 10}{5}=-2$.

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 want to find the 14th term, so $n = 14$. Substitute $a_1 = 5$, $r=-2$, and $n = 14$ into the formula:
$a_{14}=5\times(-2)^{14 - 1}$

Step3: Simplify the exponent

First, calculate the exponent: $14-1 = 13$. So we have $a_{14}=5\times(-2)^{13}$.

Step4: Calculate $(-2)^{13}$

$(-2)^{13}=-2^{13}=-8192$ (since an odd power of a negative number is negative).

Step5: Multiply by 5

Now, multiply by 5: $a_{14}=5\times(-8192)=-40960$.

Answer:

-40960