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**tsyet12** How can u use substitution U_{k-1 }= 2^{k-1 }- 1 ?

The thing we are going to proof by induction is U_{n}=2^{n}-1 . A number is substitute to check if any number fulfills the equation. And if yes, we assume that there are a set of numbers that fulfills the equation and let it be k. Hence , U_{k}=2^{k} -1 is deduced and usable in further calculations. Then we proceed to proving that "k+1" or "k-1" also fulfills the equation, and if yes, the equation is correct, proven by induction.

But, in the process of proving U_{k+1, }you substitute U_{k-1 } (which wasn't deduced or proven). Is that valid? And, isn't that assuming that the equation is already correct?

If you assume that U_{k}=2^{k}-1 and U_{k}_{-1}=2^{k-1} -1 at the same time, isn't that already assuming the equation correct?