What is the value of 2^3-1^3+4^3-3^3+6^3-5^3 +…+18^3-17^3?

2023 AMC 12A Problem 12

Attempt Problem 12 of the 2023 AMC 12A 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 2023 AMC 12A solutions, or check the answer key.

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12.

What is the value of

\begin{gathered} 2^3-1^3+4^3-3^3+6^3-5^3\\ {}+\cdots+18^3-17^3? \end{gathered}

2023

2679

2941

3159

3235

Group the terms into pairs

(2k)^3-(2k-1)^3

for

k=1

9

Each pair equals

12k^2-6k+1;

sum using

\sum k^2

and

\sum k

Group into pairs

(2k)^3-(2k-1)^3

for

k=1,\ldots,9.

Expanding,

\begin{gathered} (2k)^3-(2k-1)^3\\ {}=12k^2-6k+1. \end{gathered}

Summing for

k=1

9,

with

\sum k^2=285

and

\sum k=45,

gives

\begin{gathered} 12\cdot 285-6\cdot 45+9\\ {}=3420-270+9\\ {}=3159. \end{gathered}

Thus, the correct answer is D.