2020 AMC 12B Problem 1

Attempt Problem 1 of the 2020 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2020 AMC 12B solutions, or check the answer key.

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What is the value in simplest form of the following expression?

\begin{aligned} &\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} \\ &\quad {}+ \sqrt{1+3+5+7} \end{aligned}

5

4 + \sqrt{7} + \sqrt{10}

10

15

4 + 3\sqrt{3} + 2\sqrt{5} + \sqrt{7}

The sum of the first

k

odd numbers is

k^2

Each term simplifies to

\sqrt{1},

\sqrt{4},

\sqrt{9},

and

\sqrt{16}

The sum of the first

k

odd numbers equals

k^2,

so each radicand is a perfect square:

\begin{aligned} &\sqrt{1} + \sqrt{4} + \sqrt{9} + \sqrt{16} \\ &= 1 + 2 + 3 + 4 = 10. \end{aligned}

Thus, the correct answer is C.

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