How do you write a rule for the nth term of the geometric term given the two terms $a_{3} = 24, a_{5} = 96$ $a_3=24, a_5=96$ ?

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How do you write a rule for the nth term of the geometric term given the two terms $a_{3} = 24, a_{5} = 96$ $a_3=24, a_5=96$ ?

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Answer 1 · 2017-08-05T16:06:14

$a_{n} = 6 \cdot 2^{n - 1}$ $a_n = 6 * 2^(n-1)$ or $a_{n} = 6 \cdot {(- 2)}^{n - 1}$ $a_n = 6 * (-2)^(n-1)$

Explanation:

The general formula for a geometric sequence is $a_{n} = a_{1} \cdot r^{n - 1}$ $a_n = a_1 * r^(n-1)$ , where $a_{n}$ $a_n$ is the $n^{t h}$ $n^(th)$ term, $a_{1}$ $a_1$ is the first term, and $r$ $r$ is the common ratio.

I'm going to explain how to do this problem two ways.

The Long Way

Since we are given $a_{3} = 24$ $a_3 = 24$ and $a_{5} = 96$ $a_5 = 96$ , we can substitute them into the formula.

$a_{3} = a_{1} \cdot r^{3 - 1}$ $a_3 = a_1 * r^(3-1)$
$a_{3} = a_{1} \cdot r^{2}$ $a_3 = a_1 * r^2$
$24 = a_{1} \cdot r^{2}$ $color(blue)(24 = a_1 * r^2)$

$a_{5} = a_{1} \cdot r^{5 - 1}$ $a_5 = a_1 * r^(5-1)$
$a_{5} = a_{1} \cdot r^{4}$ $a_5 = a_1 * r^4$
$96 = a_{1} \cdot r^{4}$ $color(blue)(96= a_1 * r^4)$

Now we can solve the system of equations:

$24 = a_{1} \cdot r^{2}$ $color(blue)(24 = a_1 * r^2)$ $\to$ $->$ solve for $a_{1}$ $a_1$

$a_{1} = \frac{24}{r^{2}}$ $a_1=24/r^2$

$96 = a_{1} \cdot r^{4}$ $color(blue)(96= a_1 * r^4)$

$96 = \frac{24}{r^{2}} \cdot r^{4}$ $96=24/r^2 * r^4$ $\to$ $->$ substitute the value of $a_{1}$ $a_1$ into the second equation

$96 = 24 \cdot r^{2}$ $96=24 * r^2$

$4 = r^{2}$ $4=r^2$

$r = \pm 2$ $r=+-2$

Now that we have the value of $r$ $r$ , we can find the value of $a_{1}$ $a_1$ . Using the first equation, $24 = a_{1} \cdot r^{2}$ $color(blue)(24 = a_1 * r^2)$ , we get

$24 = a_{1} \cdot r^{2}$ $24 = a_1 * r^2$

$24 = a_{1} \cdot {(\pm 2)}^{2}$ $24 = a_1 * (+-2)^2$

$24 = a_{1} \cdot 4$ $24 = a_1 * 4$

$a_{1} = 6$ $a_1=6$

So our formula for the sequence can be either $a_{n} = 6 \cdot 2^{n - 1}$ $color(red)(a_n = 6 * 2^(n-1))$ or $a_{n} = 6 \cdot {(- 2)}^{n - 1}$ $color(red)(a_n = 6 * (-2)^(n-1))$ .

To verify if these are correct, you can write out the first few terms and see if they match the information given in the problem.

$a_{n} = 6 \cdot 2^{n - 1}$ $color(red)(a_n = 6 * 2^(n-1))$

The common ratio is $2$ $2$ , so start with $6$ $6$ and multiply each term by $2 \Rightarrow 6, 12, 24, 48, 96$ $2 => 6, 12, 24, 48, 96$

$a_{n} = 6 \cdot {(- 2)}^{n - 1}$ $color(red)(a_n = 6 * (-2)^(n-1))$

The common ratio is $- 2$ $-2$ , so start with $6$ $6$ and multiply each term by $- 2 \Rightarrow 6, - 12, 24, - 48, 96$ $-2 => 6, -12, 24, -48, 96$

In both of these formulas, $a_{3} = 24$ $a_3=24$ and $a_{5} = 96$ $a_5=96$ .

The Short Way

We are given $a_{3}$ $a_3$ and $a_{5}$ $a_5$ , so we can easily find out $a_{4}$ $a_4$ in order to get the value of $r$ $r$ .

$a_{3}, a_{4}, a_{5}$ $a_3, a_4, a_5$

$24, a_{4}, 96$ $24, a_4, 96$

To find $a_{4}$ $a_4$ , we can simply calculate the geometric mean.

${(a_{4})}_{4}^{2} = 24 \cdot 96 \Rightarrow a_{4} = \pm \sqrt{24 \cdot 96} = \pm \sqrt{2304} = \pm 48$ $(a_4)^2 = 24 * 96 => a_4 = +-sqrt(24 * 96) = +-sqrt2304 = +-48$

So the three terms are either $24, 48, 96$ $24, 48, 96$ , meaning that $r = \frac{48}{24} = 2$ $r = 48/24 = 2$ , or the terms are $24, - 48, 96$ $24, -48, 96$ , meaning that $r = - \frac{48}{24} = - 2$ $r=-48/24 = -2$ .

After you find $r$ $r$ , you can find $a_{1}$ $a_1$ the same way we did above. In the end, you get $a_{n} = 6 \cdot 2^{n - 1}$ $color(red)(a_n = 6 * 2^(n-1))$ or $a_{n} = 6 \cdot {(- 2)}^{n - 1}$ $color(red)(a_n = 6 * (-2)^(n-1))$ .

(This method is easier in the context of this problem, but if you were given terms such as $a_{10}$ $a_10$ and $a_{19}$ $a_19$ , you would definitely want to use the first method.)

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How do you write a rule for the nth term of the geometric term given the two terms $a_{3} = 24, a_{5} = 96$ $a_3=24, a_5=96$ ?

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How do you write a rule for the nth term of the geometric term given the two terms a_3=24, a_5=96a3=24,a5=96a_3=24, a_5=96?

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How do you write a rule for the nth term of the geometric term given the two terms $a_{3} = 24, a_{5} = 96$ $a_3=24, a_5=96$ ?