Use induction to prove for all n∈Nn∈N that n∑k=02k=2n+1−1.
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Let be the proposition that
for the integer
is true because
This completes the basis step.
INDUCTIVE STEP: For the inductive hypothesis, we assume that is true for an arbitrary nonnegative integer That is, we assume that
To carry out the inductive step using this assumption, we must show that when we assume that is true, then is also true. That is, we must show that
assuming the inductive hypothesis Under the assumption of we see that
Note that we used the inductive hypothesis in the second equation in this string of equalities to replace by We have completed the inductive step.
Because we have completed the basis step and the inductive step, by mathematical induction we know that is true for all nonnegative integers That is,
for all nonnegative integers