Math Problem Statement

This product is equal to: (1 + 1/9) × (1 + 1/8) × (1 + 1/7) × ... × (1 + 1/2)

Solution

The problem involves evaluating the product:

(1+19)×(1+18)×(1+17)××(1+12)\left(1 + \frac{1}{9}\right) \times \left(1 + \frac{1}{8}\right) \times \left(1 + \frac{1}{7}\right) \times \dots \times \left(1 + \frac{1}{2}\right)

Let's simplify each term in the product individually:

  1. Each term can be rewritten as: 1+1n=n+1n1 + \frac{1}{n} = \frac{n + 1}{n}

  2. Therefore, the product becomes: 109×98×87××32\frac{10}{9} \times \frac{9}{8} \times \frac{8}{7} \times \dots \times \frac{3}{2}

  3. Notice that this sequence has a telescoping effect, where most terms in the numerator and denominator will cancel each other out. Specifically:

    109×98×87××32=102=5\frac{10}{9} \times \frac{9}{8} \times \frac{8}{7} \times \dots \times \frac{3}{2} = \frac{10}{2} = 5

  4. Therefore, the result of the entire product is:

5\boxed{5}

The correct answer is (B) 5.


Would you like a further breakdown of telescoping products or have any questions on similar problems? Here are some additional questions that might expand your understanding:

  1. How does the telescoping property simplify complex products or sums?
  2. What other types of mathematical expressions benefit from telescoping?
  3. Could a similar technique apply to sums rather than products?
  4. How would this approach change if terms didn’t fully cancel?
  5. What other problems could be simplified using partial fraction decomposition?

Tip: When faced with a long sequence of fractions in a product, look for a telescoping pattern—it often simplifies calculations significantly!

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Math Problem Analysis

Mathematical Concepts

Algebra
Fractions
Telescoping Products

Formulas

1 + 1/n = (n + 1)/n

Theorems

Telescoping property of products

Suitable Grade Level

Grades 8-10