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

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

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Answer 1 · 2016-02-11T04:03:00

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

1 Answer

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