Q.Prove the following by using the principle of mathematical induction for all $n \in N$: $\dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \ldots + \dfrac{1}{2^n} = 1 - \dfrac{1}{2^n}$

Let $P(n)$ be the statement

Base case: For $n=1$,

So $P(1)$ is true.

Inductive step: Assume $P(k)$ is true for some $k\ge1$:

We must show the sum up to $(k+1)$ equals $1-\dfrac{1}{2^{k+1}}$.

Adding the next term $\dfrac{1}{2^{k+1}}$ to both sides of the hypothesis: …

Geometric-series induction. Add the next term $\frac{1}{2^{k+1}}$ to $1-\frac{1}{2^k}$, rewrite $\frac{1}{2^k}=\frac{2}{2^{k+1 …

• Not rewriting $\frac{1}{2^k}$ with denominator $2^{k+1}$ before combining fractions.
• Sign error, adding instead of subtracting the next term's contribution: the correct simplification is $-\frac{2}{2^{k+1}}+\frac{1}{2^{k+1}}=-\frac{1}{2^{k+1}}$. …