Let S=51+521+531+⋅+⋅+⋅+5n1+⋅⋅⋅
Factor out 51: S=51(1+51+521+⋅+⋅+⋅+5n−11+⋅⋅⋅).
Do you see that, after the "1+ " the sum on the right is just the original sum, with the index shifted? And since the sum is infinite, that is just the original "S"!
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