How do you find all sets of three consecutive odd integers whose sum is between 20 and 30?

1 Answer
sente
Feb 11, 2016

Write the odd numbers in the form #2k+1, 2k+3, 2k+5# and set up an inequality to find all valid values for #k# to find that the sets are
#{5, 7, 9}# and #{7, 9, 11}#

Explanation:

Any set of three consecutive odd integers may be written as
#{2k+1, 2k+3, 2k+5}# for some #k in ZZ#

Then, we just need to find a condition on #k# such that
#20<(2k+1)+(2k+3)+(2k+5)<30#

Simplifying, we get

#20<6k+9<30#

Subtracting #9# gives us

#11 < 6k < 21#

Dividing by #6#

#11/6 < k < 21/6#

Thus we will have a set with the desired property when #k# is an integer between #11/6# and #21/6#, that is, when #k=2# or #k=3#. This gives us the result that the only such sets are

#{2(2)+1, 2(2)+3, 2(2)+5} = {5, 7, 9}#
and
#{2(3)+1, 2(3)+3, 2(3)+5} = {7, 9, 11}#

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