How to prove a formula for the sum of powers of 2 by induction?

Here is where I"m getting off traông chồng. Lets look at the right side of the last equation: \$2^n+1 -1\$ I can rewrite this as the following.

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\$2^1(2^n) - 1\$ But, from our hypothesis \$2^n = 2^n+1 - 1\$ Thus:

\$2^1(2^n+1 -1) -1\$ This is where I get lost. Because when I distribute through I get this.

\$2^n+2 -2 -1\$ This is wrong is it not? Am I not applying the rules of exponents correctly here? I have the solution so I know what I"m doing is wrong. Here is the correct proof.

summation induction
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edited Mar 8 "15 at 4:14
user147263
asked Feb 18 "11 at 0:37

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Inductive Step khổng lồ prove sầu is: \$ 2^n+1 = 2^n+2 - 1\$Our hypothesis is: \$2^n = 2^n+1 -1\$

are wrong & should be

Inductive sầu Step lớn prove sầu is: \$ 2^0 + 2^1 + ... + 2^n + 2^n+1 = 2^n+2 - 1\$Our hypothesis is: \$ 2^0 + 2^1 + ... + 2^n = 2^n+1-1\$

Add \$2^n+1\$ to both sides of the hypothesis and you have the step lớn prove since \$2^n+1-1 +2^n+1 = 2^n+2 - 1\$

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answered Feb 18 "11 at 0:44

HenryHenry
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An easy way to lớn vày this is using binary. Here"s an idea of what I mean:

\$2^0\$ in binary is \$1\$\$2^1\$ in binary is \$10\$\$2^2\$ in binary is \$100\$

For a general rule:

\$2^n\$ in binary is \$100cdots0\$ (n zeros)

Add those together & you get \$2^0 + 2^1 + ... + 2^n\$ in binary is \$11...11\$ (\$n+1\$ ones).

Now it"s obvious that adding 1 to lớn that gives you\$\$100cdots00 quad ext (\$n+1\$ zeros)\$\$Which as we all know is \$2^n+1\$.

Thus \$2^n+1 - 1\$ is equal so the sum of powers of two up lớn \$2^n\$.

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edited Jan 11 "16 at 0:30

JnxF
answered Jan 11 "16 at 0:12

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HINT \$ \$ Here"s the inductive proof for summing a general geometric series.

THEOREM \$ mquad 1 + x + cdots + x^n-1 = dfracx^n-1x-1\$

Proof \$ \$ Base case: It is true for \$ m n = 1:,:\$ viz. \$ m 1 = (x-1)/(x-1):\$.

Inductive step: Suppose it is true for \$ m n = k:. \$ Then we have

\$\$ m x^k + (x^k-1 +: cdots: + 1): =: x^k +fracx^k-1x-1 = fracx^k+1-1x-1\$\$

which implies it is true for \$ m: n = k+1:,:\$ thus completing the inductive sầu proof.

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The proof you seek is just the special case \$ m x = 2 \$.

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answered Feb 18 "11 at 1:19

Bill DubuqueBill Dubuque
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I don"t see the answer I like here, so I"m writing my own.

Basic proof:

We wish to prove \$2^0 + 2^1 + ... + 2^n-1 = 2^n - 1\$ for all \$n\$. We can verify by inspection this is true for n=1. Next, assume that \$2^0 + 2^1 + ... + 2^n = 2^n+1 - 1\$.

\$(2^0 + 2^1 + ... + 2^n) + 2^n+1 = (2^n+1 - 1) + 2^n+1 = 2 cdot 2^n+1 - 1 = 2^n+2 - 1\$, so we have shown \$2^0 + 2^1 + ... + 2^n-1 = 2^n - 1\$ is true for all n.

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edited Jan 18 at 7:52
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answered Atruyền thông quảng cáo 9 "18 at 4:12
EratosthenesEratosthenes
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