Answer
$a_{10} = 10$
Work Step by Step
We are given that the common ratio of this sequence, $r$, is $-\frac{1}{2}$ and that the eleventh term, $a_11$, is $-5$. Substitute these values into the explicit formula, which is $a_n = a_1 \bullet r^{n - 1}$, to find the $1st$ term:
$-5 = a_1 \bullet (-\frac{1}{2})^{11 - 1}$
Simplify the exponent:
$-5 = a_1 \bullet (-\frac{1}{2})^{10}$
Evaluate the exponential term first:
$-5 = a_1 \bullet (-\frac{1}{1024})$
Divide both sides of the equation by $-\frac{1}{256}$:
$a_{1} = 5120$
Now that we have the values of both $a_1$ and $r$, we can now find the value of $a_{10}$ by using the explicit formula for geometric sequences again:
$a_{10} = 5120 \bullet (-\frac{1}{2})^{10 - 1}$
Simplify the exponent:
$a_{10} = 5120 \bullet (-\frac{1}{2})^{9}$
Evaluate the exponential term first:
$a_{10} = 5120 \bullet (\frac{1}{512})$
Multiply to simplify:
$a_{10} = \frac{5120}{512}$
Divide to solve:
$a_{10} = 10$
