Help please with this proving problem.

[math_learner]

 

[Deleted member 4993]

View attachment 19495

The first step of "proof by induction" is to prove that the given equation is true for at least one 'n' (usually n=0 or 1).

Please show us what you have tried and exactly where you are stuck.

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https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.

 

[math_learner]

I did the first step; I've proved the first condition with n=1. I need to prove it for n+1 right now but I can't go further.

 

[HallsofIvy]

I prefer not to use "n", the same symbol that is used in the problem, for a specific value.

Instead "assuming the statement is true for n= k, prove it is true for n= k+ 1".

For n= k the statement is
$\displaystyle 2(2^1)+ 3(2^2)+ 4(2^3)+ \cdot\cdot\cdot+ (k+1)(2^k)= k(2^{k+1})$

Assuming that is true you want to prove that the statement, when n= k+1,
$\displaystyle 2(2^1)+ 3(2^2)+ 4(2^3)+ \cdot\cdot\cdot+ (k+1)(2^{k+2}= (k+1)(2^{k+2})$.

The reason so "prove by induction" problems involve sums is that the "k+1" statement is just the "k" statement plus one additional term. Since $\displaystyle 2(2^1)+ 3(2^2)+ 4(2^3)+ \cdot\cdot\cdot+ (k+1)(2^k)= k(2^{k+1})$,

$\displaystyle 2(2^1)+ 3(2^2)+ 4(2^3)+ \cdot\cdot\cdot+ k(2^{k+1})+ (k+1)(2^{k+2})=k(2^{k+1})+ (k+1)2^{k+2}$.
We can factor $\displaystyle 2^{k+1}$ out of that: $\displaystyle 2^{k+1}(k+ (k+1)^2)$.

Can you finish that?