What's the common ratio in 5/12, 1/3, 4/15, 16/75, 64/375?

1 Answer
Dec 3, 2015

Answer:

#4/5#

Explanation:

Divide any number in this sequence by the preceding number and you get #4/5#. That's the common ratio.

For example: #1/3 div 5/12=1/3 * 12/5=4/5#,

#4/15 div 1/3=4/15 * 3=12/15=4/5#,

#16/75 div 4/15=16/75 * 15/4=4/5#, and

#64/375 div 16/75 = 64/375 * 75/16 = 4/5#.

A cool thing related to this is the corresponding geometric series (infinite "sum") #5/12+1/3+4/15+16/75+64/375+cdots#. If #a# is the first term of such a series, and #r# is the common ratio, with #|r|<1#, the series converges to #a/(1-r)#.

For this example, #a=5/12# and #r=4/5# so that the series converges to #(5/12)/(1-4/5)=(5/12)/(1/5)=5/12 * 5=25/12# and we write

#5/12+1/3+4/15+16/75+64/375+cdots=25/12#.