document.write( "Question 825050: please help me with this
\n" ); document.write( "use the method of mathematical induction to prove the following
\n" ); document.write( "1) 3^4n+2 + 2*4^3n+1 is exactly divisible by 17
\n" ); document.write( "2) 11^2n -1 is a multiple of 120
\n" ); document.write( "3) n(n+1)(n+2) is an integer multiple of 6
\n" ); document.write( "4) n summation 1 r^3= n^2(n+1)^2 all over 4
\n" ); document.write( "thanks. \n" ); document.write( "

Algebra.Com's Answer #497069 by math-vortex(648) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "Hi there--\r\n" );
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document.write( "I'll demonstrate the proof by induction method for #3. Then, you try the others by applying \r\n" );
document.write( "the same method. Email me offline of you get stuck and I'll give you some more help.\r\n" );
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document.write( "The Problem:\r\n" );
document.write( "3) Prove that n(n+1)(n+2) is an integer multiple of 6.\r\n" );
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document.write( "The Solution:\r\n" );
document.write( "We will use the proof by induction method. You did not mention this in your problem \r\n" );
document.write( "statement, but we assume that n is a positive integer, n = 1, 2, 3, and so on.\r\n" );
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document.write( "Think of an induction proof as climbing a staircase. The first step of any induction proof is to \r\n" );
document.write( "show that the statement is true for a specific step on the staircase. We will show that the \r\n" );
document.write( "statement is true when n = 1.\r\n" );
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document.write( "Let n = 1.\r\n" );
document.write( "n(n+1)(n+2) = (1)(2)(3) = 6.\r\n" );
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document.write( "We know that 6*1=6, so 6 is an integer multiple of 6. The statement is true when n = 1.  \r\n" );
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document.write( "Now we are going to show that when the statement is true for some value of n, that it is also \r\n" );
document.write( "true for the next value of n. Let's have that \"any value of n\" be k. Then the next step after \r\n" );
document.write( "n = k will be n = k+1.\r\n" );
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document.write( "This part of an induction proof is the one that is usually most confusing for folks. \r\n" );
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document.write( "We make the Induction Hypothesis: Suppose that n(n+1)(n+2) is an integer multiple of 6 \r\n" );
document.write( "when n=k. \r\n" );
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document.write( "We will show that it is an integer multiple of 6 when n = k+1.\r\n" );
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document.write( "This is not the same as the first part, where we named an actual number where the statement \r\n" );
document.write( "is true. Now we say, \"Let's assume that the statement is true, somewhere, out there in space, \r\n" );
document.write( "we're not saying where, maybe I don't even know where; just somewhere.\" \r\n" );
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document.write( "If we can prove, assuming that n(n+1)(n+2) is an integer multiple of 6 for n = k, it is also an \r\n" );
document.write( "integer multiple of 6 for n = k + 1 (that is, if it works on some step on the staircase, then it \r\n" );
document.write( "must also work at the next step on the staircase), then, since we do know of a certain place \r\n" );
document.write( "(n = 1) where the statement is true, we will have proved that it is true for all values of n.\r\n" );
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document.write( "OK, let's do it:\r\n" );
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document.write( "Assume that n(n+1)(n+2) is an integer multiple of 6 when n=k. In other words, assume \r\n" );
document.write( "that k(k+1)(k+2) is an integer multiple of 6. We will show that n(n+1)(n+2) is also an integer \r\n" );
document.write( "multiple of 6 when n=k+1. In other words, substitute k+1 for n and show that this expression \r\n" );
document.write( "is an integer multiple of 6.\r\n" );
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document.write( "Now we are going to do a bunch of algebra to get this expression into a form that we can \r\n" );
document.write( "show is an integer multiple of 6.\r\n" );
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document.write( "Multiply everything out using the distributive property.\r\n" );
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document.write( "Combine like terms.\r\n" );
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document.write( "This is where we use our Induction Hypothesis to help our proof. Notice that \r\n" );
document.write( ". Multiply is out yourself to check. Rearrange our equation to \r\n" );
document.write( "separate this polynomial.\r\n" );
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document.write( "By the Induction Hypothesis,  is also an integer multiple of 6. Then we will have \r\n" );
document.write( "an integer multiple of 6 plus an integer multiple of 6 which is also an integer multiple of 6.\r\n" );
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document.write( "Factor a 3 out of the second polynomial.\r\n" );
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document.write( "Now we have 3 * (k+1) * (k+2). Notice that (k+1) and (k+2) are consecutive integers. One of \r\n" );
document.write( "them is even and one of them is odd. By definition, the even integer is an integer \r\n" );
document.write( "multiple of 2. Thus we have three factors: 3, an integer multiple of 2 and an odd integer.  \r\n" );
document.write( "Their product must be an integer multiple of 6.\r\n" );
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document.write( "The expression (n)(n+1)(n+2) is an integer multiple of 6 when n = k+1. Therefore the \r\n" );
document.write( "expression (n)(n+1)(n+2) is an integer multiple of 6 for all positive integers n.\r\n" );
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document.write( "That's it. Now you try the other problems using these steps:\r\n" );
document.write( "1) Prove the statement is true for an initial value of n (usually n=0 or n=1)\r\n" );
document.write( "2) Make the Induction Hypothesis: assume that the statement is true for n=k.\r\n" );
document.write( "3) Show that the statement if n=k is true then n=k+1 is also true.\r\n" );
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document.write( "Feel free to email me at the address below if questions arise. Good luck!\r\n" );
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document.write( "Mrs. Figgy\r\n" );
document.write( "math.in.the.vortex@gmail.com\r\n" );
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