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I have been reading the mathematical induction and attempt to solve this given problem

Prove that $$2\cdot2^1 + 3\cdot2^2 + 4\cdot2^3 + 5\cdot2^4 +\ldots+ (n+1)2^n = n2^{n+1}\;.$$

Any clues?

Brian M. Scott
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amie
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1 Answers1

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I’ll get you started. First get the induction off the ground by checking that the formula is correct when $n=1$:

$$2\cdot2^1=4=1\cdot2^{1+1}\;.$$

Now assume as your induction hypothesis that

$$2\cdot2^1+3\cdot2^2+\ldots+(n+1)2^n=n2^{n+1}\tag{1}$$

for some $n\ge 1$ and try to show that the result is true for $n+1$, i.e., that

$$2\cdot2^1+3\cdot2^2+\ldots+(n+1)2^n+(n+2)2^{n+1}=(n+1)2^{n+2}\;.\tag{2}$$

Notice that the first $n$ terms on the lefthand side of $(2)$ are identical to the lefthand side of $(1)$, so by applying the induction hypothesis $(1)$, we can conclude that

$$2\cdot2^1+3\cdot2^2+\ldots+(n+1)2^n+(n+2)2^{n+1}=n2^{n+1}+(n+2)2^{n+1}\;.\tag{3}$$

Now do a little algebraic manipulation of the righthand side of $(3)$ to show that it’s equal to the righthand side of $(2)$, and you’ll have proved that $(2)$ holds.

Brian M. Scott
  • 616,228
  • how would you show a little algebraic manipulation of the RHS of (3) to show that it is equal to the RHS of (2)??? i am confused by the explanation you have given until the last – amie Jun 05 '13 at 02:51
  • @amie: Both terms of $n2^{n+1}+(n+2)2^{n+1}$ have a factor of $2^{n+1}$, so the first thing to try is factoring it out. If you do that, you get $(2n+2)2^{n+1}$. Can you see why that’s equal to the righthand side of $(2)$? – Brian M. Scott Jun 05 '13 at 02:55
  • frankly no, im so lost. im so sorry – amie Jun 05 '13 at 02:59
  • @amie: It’s just a little more factoring and one of the laws of exponents: $$(2n+2)2^{n+1}=2(n+1)2^{n+1}=(n+1)\left(2\cdot2^{n+1}\right)=(n+1)2^{n+2};.$$ – Brian M. Scott Jun 05 '13 at 03:01
  • okay i get it now. however, 2⋅2^1+3⋅2^2+4⋅2^3+5⋅^4+…+(n+1)2n=n2^n+1 is not equal to the final answer which is (n+1)2^2n+2. shouldnt it be equal to n2^n+1??? or am i misunderstanding your computation? – amie Jun 05 '13 at 03:06
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    @amie: It appears that the real problem is that you don’t understand how mathematical induction works. There’s a concise description of it in the first part of this answer, and my answer to this question gives you another illustration of the proof method in action. – Brian M. Scott Jun 05 '13 at 03:18