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example:

1 + 2 + 4 + 8 + 16 ........ + 2 ^ N = 2 x ( 2 ^ N ) - 1

Conceptually where does the rule come from that there is a minus at the end of the equation.

It starts with an odd number so the answer must be an odd number and that's why -1 is subtracted at the end then how come if an equation started with -5 or -8 you would not subtract the -5 or -8 in the end?

I've seen numerical explanations but they confuse me, can it be explained with words?