2020 AMC 12B Problem 1

Problem 1 of 25EasierAlgebra Number Theory

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}

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Solution

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.

Concepts: sum of first n odd numbers · perfect square · radical

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.