Calculate the first three terms and formula for the sequence an= (1-1/4) (1-1/9) ... (1-1/n^2)

To find the first three terms and the formula for the sequence \( a_n = \left(1 - \frac{1}{4}\right)\left(1 - \frac{1}{9}\right) \cdots \left(1 - \frac{1}{n^2}\right) \), let's break this down.

### Step 1: Simplifying each term

Each factor in the product takes the form:
\[
1 - \frac{1}{k^2} = \frac{k^2 - 1}{k^2} = \frac{(k-1)(k+1)}{k^2}
\]
for each \( k \) from 2 to \( n \).

### Step 2: Writing the product for \( a_n \)

So, the sequence \( a_n \) can be written as:
\[
a_n = \prod_{k=2}^{n} \left( 1 - \frac{1}{k^2} \right) = \prod_{k=2}^{n} \frac{(k-1)(k+1)}{k^2}
\]

### Step 3: Calculating the first three terms

1. **For \( n = 2 \):**
   \[
   a_2 = \left(1 - \frac{1}{4}\right) = \frac{3}{4}
   \]

2. **For \( n = 3 \):**
   \[
   a_3 = \left(1 - \frac{1}{4}\right)\left(1 - \frac{1}{9}\right) = \frac{3}{4} \times \frac{8}{9} = \frac{24}{36} = \frac{2}{3}
   \]

3. **For \( n = 4 \):**
   \[
   a_4 = \left(1 - \frac{1}{4}\right)\left(1 - \frac{1}{9}\right)\left(1 - \frac{1}{16}\right) = \frac{3}{4} \times \frac{8}{9} \times \frac{15}{16} = \frac{360}{576} = \frac{5}{8}
   \]

### Step 4: General Formula for \( a_n \)

After simplification, the sequence \( a_n \) can be expressed as:
\[
a_n = \frac{2}{n+1}
\]

### Summary:
- First three terms: 
  \[
  a_2 = \frac{3}{4}, \quad a_3 = \frac{2}{3}, \quad a_4 = \frac{5}{8}
  \]
- General formula: 
  \[
  a_n = \frac{2}{n+1}
  \]

Would you like more details or have any further questions?

Here are five related questions:
1. How would the formula for \( a_n \) change if the product started from \( k=1 \) instead of \( k=2 \)?
2. Can you find the limit of \( a_n \) as \( n \to \infty \)?
3. What is the sum of the sequence \( a_n \) from \( n=2 \) to infinity?
4. How would the sequence change if each term was raised to a power?
5. Can the sequence be expressed as a continued fraction?

**Tip:** Recognizing patterns in product sequences often leads to telescoping series, which simplify the process of finding general formulas.

Learn how to calculate the first three terms and the general formula for the sequence an = (1 - 1/4)(1 - 1/9)... This solution walks through simplifying the terms and provides the formula an = 2/(n+1), suitable for high school students in Grades 9-12.

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