math_learner
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Help please with this proving problem.
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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.
Please follow the rules of posting in this forum, as enunciated at:
https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520
Please share your work/thoughts about this assignment.
math_learner
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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
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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?
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?
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