How to prove this mathematical induction?

Question

Can someone explain this prove mathematical induction question?

Use mathematical induction to prove the following expression: $$ \sum_{k = 1}^{n} 2^{n + 1} - 1. $$ I tried my best to solve it but when i tried to prove for p(1) it got failed and for p(0) it is working fine.

I am new to mathematical induction and have test tomorrow.

I am new to this maths stack exchange and please for give if there is any formatting mistakes.

Tried to solve this

Tried to solve this - Continued

Please help me out and thanks.

Answer 1

There is a typo in the text, it should be

$$\sum_{i=0}^n2^i=2^{n+1}-1$$

then proceed firstly by base case and then by induction step assuming true $P(n)$ and deriving from it $P(n+1)$.

Refer also to the related

• Mathematical induction (theory)
• Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction