Question 20 of 20advanced🔍 AnalyzeNumerical5 marks

What is 1−121 - \frac{1}{2}? (1−12)×(1−13)\left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{3}\right)? (1−12)×(1−13)×(1−14)×(1−15)\left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{3}\right) \times \left(1 - \frac{1}{4}\right) \times \left(1 - \frac{1}{5}\right)? (1−12)×(1−13)×(1−14)×(1−15)×(1−16)×(1−17)×(1−18)×(1−19)×(1−110)\left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{3}\right) \times \left(1 - \frac{1}{4}\right) \times \left(1 - \frac{1}{5}\right) \times \left(1 - \frac{1}{6}\right) \times \left(1 - \frac{1}{7}\right) \times \left(1 - \frac{1}{8}\right) \times \left(1 - \frac{1}{9}\right) \times \left(1 - \frac{1}{10}\right)? Make a general statement and explain.

(i)
Answer

12\frac{1}{2}

(ii)

What is (1−12)×(1−13)\left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{3}\right)?

Answer

13\frac{1}{3}

(iii)

What is the product up to (1−15)\left(1 - \frac{1}{5}\right)?

Answer

15\frac{1}{5}

(iv)

What is the product up to (1−110)\left(1 - \frac{1}{10}\right)?

Answer

110\frac{1}{10}

(v)
Answer

The product (1−12)×(1−13)×...×(1−1n)=1n\left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{3}\right) \times ... \times \left(1 - \frac{1}{n}\right) = \frac{1}{n}

Explanation

(i) 1−12=121 - \frac{1}{2} = \frac{1}{2} (ii) 12×23=13\frac{1}{2} \times \frac{2}{3} = \frac{1}{3} (iii) 12×23×34×45=15\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \frac{4}{5} = \frac{1}{5} (iv) All intermediate terms cancel, leaving 110\frac{1}{10} (v) Each term equals k−1k\frac{k-1}{k}. Multiplying all terms causes cancellation of all intermediate numerators and denominators, leaving only 1n\frac{1}{n}.

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