the sum of the first 9 term of a geometric sequence is 10220 and the sum to infinity of the sequence is 10240. find the 1st term and the common ration of the sequence

To solve for the first term \( a \) and the common ratio \( r \) of a geometric sequence, we are given the following information:

1. The sum of the first 9 terms of the sequence is 10220.
2. The sum to infinity of the sequence is 10240.

The sum of the first \( n \) terms of a geometric sequence is given by:
\[ S_n = a \frac{1-r^n}{1-r} \]

The sum to infinity of a geometric sequence, provided \( |r| < 1 \), is:
\[ S_\infty = \frac{a}{1-r} \]

From the given information:
\[ S_9 = a \frac{1-r^9}{1-r} = 10220 \]
\[ S_\infty = \frac{a}{1-r} = 10240 \]

We can use the second equation to express \( a \) in terms of \( r \):
\[ \frac{a}{1-r} = 10240 \]
\[ a = 10240 (1-r) \]

Substitute this expression for \( a \) into the first equation:
\[ 10240 (1-r) \frac{1-r^9}{1-r} = 10220 \]
\[ 10240 (1-r^9) = 10220 \]
\[ 10240 - 10240 r^9 = 10220 \]
\[ 10240 - 10220 = 10240 r^9 \]
\[ 20 = 10240 r^9 \]
\[ r^9 = \frac{20}{10240} \]
\[ r^9 = \frac{1}{512} \]
\[ r = \left( \frac{1}{512} \right)^{\frac{1}{9}} \]
\[ r = \left( 2^{-9} \right)^{\frac{1}{9}} \]
\[ r = 2^{-1} \]
\[ r = \frac{1}{2} \]

Now, we substitute \( r \) back into the equation for \( a \):
\[ a = 10240 (1 - \frac{1}{2}) \]
\[ a = 10240 \cdot \frac{1}{2} \]
\[ a = 5120 \]

Thus, the first term \( a \) is 5120 and the common ratio \( r \) is \( \frac{1}{2} \).

**Summary:**
- First term \( a = 5120 \)
- Common ratio \( r = \frac{1}{2} \)

Would you like details or have any questions?

Here are some related questions to consider:
1. How would you find the tenth term of this geometric sequence?
2. What is the general formula for the \( n \)-th term of a geometric sequence?
3. How does the common ratio affect the convergence of the sum to infinity in a geometric series?
4. Can a geometric sequence with a common ratio greater than 1 have a sum to infinity?
5. What are the properties of a geometric sequence with a common ratio between 0 and 1?
6. How do you find the sum of the first \( n \) terms if the common ratio is negative?
7. What are the differences between arithmetic and geometric sequences?
8. How does changing the first term affect the sums of the terms in a geometric sequence?

**Tip:** In geometric sequences, the common ratio determines the rate at which the terms grow or shrink. If \( |r| < 1 \), the terms decrease, and if \( |r| > 1 \), the terms increase exponentially.

Learn how to find the first term and common ratio of a geometric sequence given the sum of the first 9 terms and the sum to infinity. Detailed step-by-step solution covering geometric sequences, infinite series, and exponential growth concepts. Suitable for high school students studying sequences and series.

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