
What is the first 3 terms and formula of an=(1-1/4)(1-1/9)...(1-1/n^2) for n>=2

Let's analyze the given sequence \( a_n \) defined as:

\[
a_n = \left(1 - \frac{1}{4}\right) \left(1 - \frac{1}{9}\right) \dots \left(1 - \frac{1}{n^2}\right)
\]

This expression can be simplified by rewriting each factor \( \left(1 - \frac{1}{k^2}\right) \) as:

\[
1 - \frac{1}{k^2} = \frac{k^2 - 1}{k^2} = \frac{(k-1)(k+1)}{k^2}
\]

Hence, the product form becomes:

\[
a_n = \prod_{k=2}^{n} \frac{(k-1)(k+1)}{k^2}
\]

Now, simplifying this product:

\[
a_n = \frac{(1 \cdot 3)}{2^2} \cdot \frac{(2 \cdot 4)}{3^2} \cdot \frac{(3 \cdot 5)}{4^2} \dots \frac{(n-1)(n+1)}{n^2}
\]

Many terms will cancel out, leaving us with:

\[
a_n = \frac{1 \cdot 3}{n(n+1)}
\]

Thus, the general formula for \( a_n \) is:

\[
a_n = \frac{3}{n(n+1)}
\]

### First 3 terms of the sequence \( a_n \):

1. For \( n = 2 \):

\[
a_2 = \frac{3}{2(3)} = \frac{1}{2}
\]

2. For \( n = 3 \):

\[
a_3 = \frac{3}{3(4)} = \frac{1}{4}
\]

3. For \( n = 4 \):

\[
a_4 = \frac{3}{4(5)} = \frac{3}{20}
\]

### Summary:
- General formula: \( a_n = \frac{3}{n(n+1)} \)
- First 3 terms: \( a_2 = \frac{1}{2}, a_3 = \frac{1}{4}, a_4 = \frac{3}{20} \)

Do you want more details or have any questions? Here are 5 related questions to explore further:

1. How does the sequence behave as \( n \to \infty \)?
2. Can you find a recurrence relation for \( a_n \)?
3. How does \( a_n \) relate to the harmonic series?
4. What is the sum of the sequence \( a_n \) for finite \( n \)?
5. How can you express the product form of the sequence without cancellation?

**Tip:** Simplifying complex product forms often involves recognizing cancellation patterns in consecutive terms!

What is the first 3 terms and formula of an = (1 - 1/4)(1 - 1/9)...(1 - 1/n^2) for n >= 2?

Explore the formula and first 3 terms of the product series defined as an = (1 - 1/4)(1 - 1/9)...(1 - 1/n^2) for n >= 2. This problem involves algebraic manipulation and simplification of product forms. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation