Question

If \( r, s, t \) are consecutive odd integers with \( r<s<t \), which of the following must be true?

• A. \( r + t = 2s \)
• B. \( r + t = 2s + 2 \)
• C. \( r + s = t - 2 \)
• D. \( r + t = 2s + 5 \)

Hint:

For any three consecutive odd numbers, if the middle one is \(x\), the others will be \(x-2\) and \(x+2\). This symmetry makes the sum of the first and third equal to twice the middle.

Answer 1

The Correct Option is A

Let us assume the three consecutive odd integers are defined as follows:
\( r = x - 2 \)
\( s = x \)
\( t = x + 2 \)
\medskip Now calculate \( r + t \):
\( r + t = (x - 2) + (x + 2) = 2x \)
Now calculate \( 2s \):
\( 2s = 2x \)
Thus,
\( r + t = 2s \)
This satisfies option (A), and none of the other options fit this identity. Hence, the correct answer is:
\[ {r + t = 2s} \]