We are given the infinite geometric series:
$512 - 256 + 128 - 64 + \cdots$

Let's address each part of the question step by step:

(a) Find the common ratio

In a geometric series, the common ratio $r$ is the factor by which each term is multiplied to get the next term.

To find the common ratio: $r = \frac{\text{second term}}{\text{first term}} = \frac{-256}{512} = -\frac{1}{2}$

So, the common ratio is: $r = -\frac{1}{2}$

(b) Find the twelfth term of this series

The general term of a geometric series is given by: $T_n = a \cdot r^{n-1}$ where:

• $a = 512$ (the first term),
• $r = -\frac{1}{2}$,
• $n = 12$ (we need the 12th term).

Substituting into the formula: $T_{12} = 512 \cdot \left( -\frac{1}{2} \right)^{12-1} = 512 \cdot \left( -\frac{1}{2} \right)^{11}$ Since $(-1)^{11} = -1$, this becomes: $T_{12} = 512 \cdot -\frac{1}{2^{11}} = 512 \cdot -\frac{1}{2048}$ Simplifying: $T_{12} = -\frac{512}{2048} = -\frac{1}{4}$

So, the twelfth term is: $T_{12} = -\frac{1}{4}$

(c) Find the exact value of the sum of the infinite series

The sum of an infinite geometric series is given by the formula: $S_{\infty} = \frac{a}{1 - r}$ where $|r| < 1$.

Substituting the values $a = 512$ and $r = -\frac{1}{2}$: $S_{\infty} = \frac{512}{1 - \left( -\frac{1}{2} \right)} = \frac{512}{1 + \frac{1}{2}} = \frac{512}{\frac{3}{2}} = 512 \cdot \frac{2}{3}$ Simplifying: $S_{\infty} = \frac{1024}{3}$

Thus, the exact value of the sum of the infinite series is: $S_{\infty} = \frac{1024}{3}$

Final Answers:

(a) The common ratio is $-\frac{1}{2}$.
(b) The twelfth term is $-\frac{1}{4}$.
(c) The sum of the infinite series is $\frac{1024}{3}$.

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Do you have any questions or would you like more details on any part? Here are 5 related questions you can explore:

1. How does the common ratio affect the convergence of an infinite geometric series?
2. How would you find the sum of the first 12 terms of this series?
3. What happens to the sum of the series if the common ratio is greater than 1?
4. How can the formula for the sum of an infinite geometric series be derived?
5. What is the difference between a finite and infinite geometric series in terms of summation?

Tip: Always check that $|r| < 1$ before applying the sum formula for infinite geometric series, as this condition ensures convergence!