Consider the infinite geometric series 512-256 + 128 - 64 +•..
(a) Find the common ratio, giving your answer as a fraction in its simplest form.
(b) Find the twelfth term of this series.
(c) Find the exact value of the sum of the infinite series.

Let's solve the problem step by step:

### (a) Find the common ratio.

In a geometric series, the common ratio \( r \) is the factor by which we multiply each term to get the next term. Given the series:

\[
512 - 256 + 128 - 64 + \dots
\]

We can find the common ratio by dividing the second term by the first term:

\[
r = \frac{-256}{512} = -\frac{1}{2}
\]

So, the common ratio is:

\[
r = -\frac{1}{2}
\]

### (b) Find the twelfth term of the series.

The general term of a geometric series can be written as:

\[
T_n = a \cdot r^{n-1}
\]

Where:
- \( a = 512 \) (the first term),
- \( r = -\frac{1}{2} \),
- \( n = 12 \).

So, the twelfth term is:

\[
T_{12} = 512 \cdot \left(-\frac{1}{2}\right)^{12-1} = 512 \cdot \left(-\frac{1}{2}\right)^{11}
\]

Calculating:

\[
T_{12} = 512 \cdot \left(-\frac{1}{2}\right)^{11} = 512 \cdot -\frac{1}{2048}
\]

\[
T_{12} = \frac{-512}{2048} = -\frac{1}{4}
\]

So, the twelfth term is:

\[
T_{12} = -\frac{1}{4}
\]

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

The sum \( S \) of an infinite geometric series is given by the formula:

\[
S = \frac{a}{1 - r}
\]

Where:
- \( a = 512 \),
- \( r = -\frac{1}{2} \).

Substitute the values:

\[
S = \frac{512}{1 - \left(-\frac{1}{2}\right)} = \frac{512}{1 + \frac{1}{2}} = \frac{512}{\frac{3}{2}} = 512 \cdot \frac{2}{3} = \frac{1024}{3}
\]

So, the exact value of the sum of the infinite series is:

\[
S = \frac{1024}{3}
\]

### Final Answers:
1. **(a)** The common ratio is \( r = -\frac{1}{2} \).
2. **(b)** The twelfth term is \( T_{12} = -\frac{1}{4} \).
3. **(c)** The exact value of the sum of the infinite series is \( S = \frac{1024}{3} \).

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Consider the infinite geometric series 512 - 256 + 128 - 64 + •.. (a) Find the common ratio, giving your answer as a fraction in its simplest form. (b) Find the twelfth term of this series. (c) Find the exact value of the sum of the infinite series.

Learn how to solve the infinite geometric series 512 - 256 + 128 - 64 + ... by finding the common ratio, twelfth term, and exact sum. This detailed solution covers key geometric series formulas and is suitable for students in Grades 10-12.

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