Proving $\sum\limits_{i=0}^n i 2^{i-1} = (n+1) 2^n - 1$ by induction

I'm trying to apply an induction proof to show that $((n+1) 2^n - 1 $ is the sum of $(i 2^{i-1})$ from $0$ to $n$.

• the base case: L.H.S = R.H.S
• we assume that $(k+1) 2^k - 1 $ is true.
• we need to prove that $(k+2) 2^{k+1} - 1$

My try to prove 3 is as follows:

$(k+2) 2^{k+1} - 1$

$(k+2) (2^k * 2) - 1$ , from 2: $2^k = 1/(k+1)$

$(k+2) (2 / (k+1)) - 1$

$(k+1) 2 - 1$

My question is, how could I get to $2^k$ from the last line to prove this formula is right?

Solution 1:

You have a typo in your statement. You want to show $$ \sum_{i=0}^n i 2^{i-1} = n 2^n - 2^n + 1 $$

Base case $n = 0$:

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

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

Then make an induction step from $n$ to $n+1$. This means you want to show that given your assumption you can show

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

Do this as follows:

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

Which is what you wanted to show. Hope this helps.

Solution 2:

HINT $\ $ The RHS should be $\rm\:(n-1)\ 2^n + 1\:.\:$ As always, by telescopy, the inductive step arises from equating the first difference of the LHS and RHS. Here that yields the trivially proved identity

$$\rm (n+1)\ 2^n\ =\ n\ 2^{n+1} - (n-1)\ 2^n $$

which, combined with the trivial proof of the base case $\rm\:n=0\:,\:$ completes the proof by induction.

REMARK $\ $ Note that absolutely no ingenuity is required. The proof by telescopy is so mechanical that it can be done by a machine. Indeed, cancelling the factor of $\rm\:2^n\:,\:$ said inductive step reduces to proving the polynomial identity $\rm\ n+1\ =\ 2\:n - (n-1)\:.\:$ For much further discussion see my many posts on telescopy.