1/3 * 2/4 * 3/5 * 4/6 * 5/7 *... ... ... * 98/100 = ?
Is there a specific way on how to calculate this?
1/3 * 2/4 * 3/5 * 4/6 * 5/7 *... ... ... * 98/100
= (1*2*3*4*5*...*98)/(3*4*5*6*...*100)
notice all factors in the top from 3*4*..*98 cancel with those at the bottom, leaving you with
1*2/(99*100)
= 2/9900
= 1/4950
Yes, there is a specific way to calculate the expression 1/3 * 2/4 * 3/5 * 4/6 * 5/7 * ... * 98/100. This is a product of fractions in which each numerator and denominator increment by 1 as you go from one term to the next.
To calculate this expression, you can simplify and cancel out terms as much as possible.
Let's start solving this step by step:
Step 1: Cancel out common factors between the numerator and denominator of each fraction:
1/3 * 2/4 * 3/5 * 4/6 * 5/7 * ... * 98/100 = (1/1) * (1/2) * (3/5) * (2/3) * (5/7) * ... * (49/50) * (98/100)
In this step, we cancel out the common factors between numerator and denominator. For example, 2 in the numerator of the second term cancels out with 2 in the denominator of the first term.
Step 2: Notice that most of the fractions have a numerator that is one greater than the denominator. We can simplify it further:
(1/1) * (1/2) * (3/5) * (2/3) * (5/7) * ... * (49/50) * (98/100) = (1/1) * (1/2) * (3/2) * (2/3) * (5/3) * (49/50) * (98/100)
Here, we simplified each fraction by multiplying the numerator and denominator by the reciprocal of the numerator.
Step 3: Cancel out terms that appear both in the numerator and denominator:
(1/1) * (1/2) * (3/2) * (2/3) * (5/3) * (49/50) * (98/100) = (1/1) * (1/1) * (1/3) * (1/1) * (5/1) * (49/50) * (1/1)
In this step, we can cancel out terms where numerator and denominator are the same.
Step 4: Multiply the numerators and the denominators:
(1/1) * (1/1) * (1/3) * (1/1) * (5/1) * (49/50) * (1/1) = 1/30 * 245/50
Step 5: Simplify the fraction:
1/30 * 245/50 = 1/6 * 7/2 = 7/12
Therefore, 1/3 * 2/4 * 3/5 * 4/6 * 5/7 * ... * 98/100 equals 7/12.