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.

Understand the Problem

The question is asking to derive an expression for the terms of a geometric sequence given that the third term is 1000 and the common ratio is -1/5. Then, it requires us to calculate the sixth term of the sequence, g6.

Answer

$g_6 = -8$
Answer for screen readers

The sixth term of the geometric sequence, $g_6$, is $-8$.

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.

  1. 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:

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

  1. 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:

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

  1. Solve for the first term $g_1$

Multiply both sides by 25:

$$ g_1 = 1000 \times 25 $$

Calculating gives:

$$ g_1 = 25000 $$

  1. 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:

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

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

Calculating this term gives:

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

  1. 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:

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

  1. Simplify the fraction

Evaluating this gives:

$$ g_6 = -8 $$

The sixth term of the geometric sequence, $g_6$, is $-8$.

More Information

The geometric sequence is characterized by a constant ratio between successive terms. In this case, the common ratio is negative, which means the terms alternate in sign. Geometric sequences can model various real-world phenomena, including exponential growth or decay.

Tips

  • Miscalculating the power of the common ratio.
  • Forgetting to alternate signs for terms with a negative common ratio.
  • Mixing up the positions of the terms in the general formula.
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