Track / Stage 3 / 32 of 260 #32 of 2000

Problem 32

AMC 10/12, early questions

Number theory Difficulty 3.1 Find the answer

Observe the following: $1 \times 3 = 3$ and $3 = 2^2 - 1$ , $3 \times 5 = 15$ and $15 = 4^2 - 1$ , $5 \times 7 = 35$ and $35 = 6^2 - 1$ , ..., $11 \times 13 = 143$ and $143 = 12^2 - 1$ . Express the pattern you guess in a formula containing only one variable $n$ .

Your answer

Official solution

Given that $1 \times 3 = 3$ and $3 = 2^2 - 1$ ,
$3 \times 5 = 15$ and $15 = 4^2 - 1$ ,
$5 \times 7 = 35$ and $35 = 6^2 - 1$ , ..., $11 \times 13 = 143$ ,
we can observe a pattern where the second factor on the left side of the equation is 2 more than the first factor.
When this number is $n$ , the left side of the equation is $n(n + 2)$ , and the right side of the equation is $(n + 1)^2 - 1$ .
Thus, $n(n + 2) = (n + 1)^2 - 1$ .
Therefore, the answer is: $\boxed{n(n + 2) = (n + 1)^2 - 1}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.