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Solution:(x^1000 - 1)/(x - 1) = (x^999 + x^998 + ... + x + 1)(x - 1)/(x - 1)

= (x^999 + x^998 + ... + x + 1)

Substituting x = 1 I get...

Line (1):1 + 1

_{1}+ 1

_{2}+ ... + 1

_{999}= 1 + 999 = 1000

Is there any way to prove Line (1): I know it is obvious but I want to prove it mathematically and without obviously counting it on my fingers.