P is the product, indicated above, of all the numbers of the form 1 – \frac{1}{k}, where k is an integer from 2 to 50, inclusive. What is the value of P ?

Question

asked Jul 14, 2020 106k views

1 Answer

JSimonsen · Answer 1 · 2020-07-19T19:23:56+0000

$P=\displaystyle\prod_(k=2)^(50)\left(1-\frac1k\right)$

Write the
-th number in the product as

$1-\frac1k=\frac kk-\frac1k=\frac{k-1}k$

Then

$P=\frac{2-1}2\cdot\frac{3-1}3\cdot\frac{4-1}4\cdot\cdots\cdot(49-1)/(49)\cdot(50-1)/(50)$

$P=\frac12\cdot\frac23\cdot\frac34\cdot\cdots\cdot(48)/(49)\cdot(49)/(50)$

Notice how the denominator of one term cancels with the numerator of the next, leaving us with

$\boxed{P=\frac1{50}}$