Discussion
Find the third proportion to \((a^{2}+2a+1)\) and \((a+1)\).
Answer: B
The third proportion is 1.
The third proportional to a and b is b2/a. Here a = a2 + 2a + 1 = (a + 1)2 and b = a + 1.
\[ \frac{(a+1)^{2}}{(a+1)^{2}}=1 \]
(a − 1) and the two fractions with (a − 1) do not equal this value.
The third proportion c of a and b satisfies a : b = b : c, so c = b2 / a. Since a is a perfect square, (a + 1)2, the fraction cancels completely and gives exactly 1. This is the usual shortcut for such problems.
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