What is the value of the product (1322)(2433)(3544)(97999898)(981009999)?

We rewrite: 12(3223)(4334)(99989899)10099 The middle terms cancel, leaving us with (1100299)= 5099

jmc

algebra junior

Problem

What is the value of the product

(\frac{1 \cdot 3}{2 \cdot 2}) (\frac{2 \cdot 4}{3 \cdot 3}) (\frac{3 \cdot 5}{4 \cdot 4}) \dots (\frac{97 \cdot 99}{98 \cdot 98}) (\frac{98 \cdot 100}{99 \cdot 99}) ?

(A)

\frac{1}{2}

(B)

\frac{50}{99}

(C)

\frac{9800}{9801}

(D)

\frac{100}{99}

Solution

We rewrite:

\frac{1}{2} \cdot (\frac{3 \cdot 2}{2 \cdot 3}) (\frac{4 \cdot 3}{3 \cdot 4}) \dots (\frac{99 \cdot 98}{98 \cdot 99}) \cdot \frac{100}{99}

The middle terms cancel, leaving us with

(\frac{1 \cdot 100}{2 \cdot 99}) = \frac{50}{99}

Final answer