What is the sum of the first 10 term of 4,2,1,...? Show solution

The given sequence is: 4, 2, 1, ...

This is a **geometric sequence** where each term after the first is found by multiplying the previous term by a constant ratio. To determine the sum of the first 10 terms, let's first identify the relevant values:

1. **First term \(a_1\)**: The first term is \(4\).
2. **Common ratio \(r\)**: The ratio between the terms is \( \frac{2}{4} = \frac{1}{2} \).

### Formula for the sum of the first \(n\) terms of a geometric sequence:
\[
S_n = \frac{a_1(1 - r^n)}{1 - r}, \quad \text{for} \quad r \neq 1
\]
Where:
- \( S_n \) is the sum of the first \(n\) terms,
- \( a_1 \) is the first term,
- \( r \) is the common ratio, and
- \( n \) is the number of terms.

### Plugging in the values:
- \( a_1 = 4 \),
- \( r = \frac{1}{2} \),
- \( n = 10 \).

The sum of the first 10 terms:
\[
S_{10} = \frac{4(1 - \left(\frac{1}{2}\right)^{10})}{1 - \frac{1}{2}}
\]

### Simplifying:
1. \( 1 - \left(\frac{1}{2}\right)^{10} = 1 - \frac{1}{1024} = \frac{1023}{1024} \),
2. \( 1 - \frac{1}{2} = \frac{1}{2} \).

Now substitute these into the formula:
\[
S_{10} = \frac{4 \cdot \frac{1023}{1024}}{\frac{1}{2}} = 4 \cdot \frac{1023}{1024} \cdot 2 = \frac{8 \cdot 1023}{1024} = \frac{8184}{1024} = 7.9921875
\]

Thus, the sum of the first 10 terms is approximately:
\[
S_{10} \approx 7.99
\]

### Would you like further details or clarifications?

Here are some related questions:
1. How do we determine the common ratio of a geometric sequence?
2. What is the sum formula for an arithmetic series?
3. How do you calculate the sum of infinite terms in a geometric series?
4. Can the sum formula be used if the ratio is greater than 1?
5. What happens if the ratio is negative in a geometric series?

**Tip:** Always check the ratio in geometric sequences to ensure the sum formula is applicable!

What is the sum of the first 10 terms of 4, 2, 1,...? Show solution.

Find the sum of the first 10 terms of the geometric sequence 4, 2, 1,... using the geometric series sum formula. This detailed solution covers key concepts such as identifying the first term, common ratio, and applying the series sum formula. Suitable for high school students studying sequences and series.

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