Let gn be a geometric sequence with the following properties. Using an appropriate model, find an expression for gn and find g6. g3=1000, and r=-1/5.

Steps to Solve

1. Define the geometric sequence

In a geometric sequence, each term is found by multiplying the previous term by a common ratio, $r$. The general term can be expressed as:

$$ g_n = g_1 \times r^{n-1} $$

where $g_n$ is the $n$-th term, $g_1$ is the first term, and $r$ is the common ratio.

2. Write the expression for the third term

Since we know the third term ($g_3$) is 1000 and the common ratio ($r$) is $-\frac{1}{5}$, we can write:

$$ g_3 = g_1 \times r^{2} $$

Substituting the known values:

$$ 1000 = g_1 \times \left(-\frac{1}{5}\right)^{2} $$

3. Calculate $( -\frac{1}{5} )^{2}$

Calculating the square of the common ratio:

$$ \left(-\frac{1}{5}\right)^{2} = \frac{1}{25} $$

So we rewrite the equation as:

$$ 1000 = g_1 \times \frac{1}{25} $$

4. Solve for the first term $g_1$

Multiply both sides by 25:

$$ g_1 = 1000 \times 25 $$

Calculating gives:

$$ g_1 = 25000 $$

5. Use $g_1$ to find the sixth term $g_6$

Now that we have $g_1$, we can find the sixth term using the geometric sequence formula:

$$ g_6 = g_1 \times r^{5} $$

Substituting in the values we found:

$$ g_6 = 25000 \times \left(-\frac{1}{5}\right)^{5} $$

6. Calculate $\left(-\frac{1}{5}\right)^{5}$

Calculating this term gives:

$$ \left(-\frac{1}{5}\right)^{5} = -\frac{1}{3125} $$

7. Final calculation for $g_6$

Substituting this back into the equation for $g_6$:

$$ g_6 = 25000 \times \left(-\frac{1}{3125}\right) $$

Calculating gives:

$$ g_6 = \frac{25000}{-3125} $$

8. Simplify the fraction

Evaluating this gives:

$$ g_6 = -8 $$