expand
- (2n + 7)^2 = (2n + 7)(2n + 7) = 4n^2 + 28n + 49
- (2n - 7)^2 = (2n - 7)(2n - 7) = 4n^2 - 28n + 49
perform operations
- (4n^2 + 28n + 49) - (4n^2 - 28n + 49)
- 4n^2 - 4n^2 + 28n + 28n +49 - 49
- 56n
factor out 8 from the above expression
since ,56n can be written as a product of 8 and 7n which is an integer for all the positive values of n , therefore, (2n + 7)² - (2n-7)² is indeed a multiple of 8 for all positive integer values of n.
