4.1.6 n² + n = (n + 1)² − (n + 1)

This identity allows you to quickly compute expressions like $n^2 + n$ when $n$ is one away from a nice number.

The Identity

$$n^2 + n = n^2 + 2n + 1 - n - 1 = (n + 1)^2 - (n + 1)$$

Example: 89² + 89

Method	Calculation	Result
Using identity	90² − 90	8100 − 90 = 8010

Reverse Application: 66² − 66

Method	Calculation	Result
Using identity	65² + 65	4225 + 65 = 4290

Tip: This trick is especially useful when the number is one away from a multiple of 10 or 5, making both the squaring and the addition/subtraction easy!

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Problem Set 4.1.6

Practice these problems. Type your answer and press Enter to check: