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There are two parts to mathematical induction:
- Show that the formula works for 1 term. In other words. show that the formula works for n = 1.
- Assuming that the formula for n is true, show that the formula for (n+1) is true. This is done by:
- Take the formula for n and use regular Algebra to transform the left side to match what the formula would be for (n+1). This often requires that equivalent changes are made to the right side of the formula.
- Transform the right side, without touching the left side, so that it matches the pattern on right side for (n+1).
\n" ); document.write( "The first part is usually easy. For your formula, the first term is 8. Is the right side equal to 8 when n = 1? You should be able to find easily that the answer to this is \"yes\".
\n" ); document.write( "The second part usually takes the most work. First let's look at the left side of the formula for n:
\n" ); document.write( "8++16+...+8n
\n" ); document.write( "and look at the left side for (n+1):
\n" ); document.write( "8++16+...+8n+8(n+1)
\n" ); document.write( "The difference between the two, as we can see, is an extra term: 8(n+1). So we can take the formula for n:
\n" ); document.write( "8++16+...+8n = 4n(n+1)
\n" ); document.write( "and change the left side into the formula for (n+1) by adding 8(n+1) to each side:
\n" ); document.write( "8++16+...+8n+8(n+1) = 4n(n+1)+8(n+1)
\n" ); document.write( "Now we see if we can take this right side and, without changing the left side, change it into the formula for (n+1). It will help if we look ahead at what our goal is. The right side of the formula for n is: 4n(n+1). The right side of the formula for (n+1) will be what we get when we replace each n by (n+1):
\n" ); document.write( "4(n+1)((n+1)+1)
\n" ); document.write( "So our task now is to try to transform the right side we have now
\n" ); document.write( "4n(n+1)+8(n+1)
\n" ); document.write( "into
\n" ); document.write( "4(n+1)((n+1)+1)
\n" ); document.write( "without changing the left side of the equation.
\n" ); document.write( "There are probably many ways to do this. I notice that (n+1) is a factor of both what we have, 4n(n+1)+8(n+1), and what we want, 4(n+1)((n+1)+1), so I will start by factoring out (n+1) from 4n(n+1)+8(n+1) giving us:
\n" ); document.write( "8++16+...+8n+8(n+1) = (n+1)(4n+8)
\n" ); document.write( "I also notice that 4 is a factor of our goal. So I will factor out a 4 out of the (4n+8):
\n" ); document.write( "8++16+...+8n+8(n+1) = 4(n+1)(n+2)
\n" ); document.write( "The last step should be a bit obvious. We have the factors of 4 and (n+1) of our goal. The third factor of what we have is (n+2) and we want it to be ((n+1)+1). Since 2 = 1+1 we can change the (n+2) into (n+1+1). And since addition is associative, we can group it any way we want. So (n+2) = (n+1+1) = ((n+1)+1). Now we have:
\n" ); document.write( "8++16+...+8n+8(n+1) = 4(n+1)((n+1)+1)
\n" ); document.write( "In summary, we took the formula for n:
\n" ); document.write( "8++16+...+8n = 4n(n+1)
\n" ); document.write( "and, using regular Algebra, we changed it into the formula for (n+1):
\n" ); document.write( "8++16+...+8n+8(n+1) = 4(n+1)((n+1)+1)
\n" ); document.write( "This, combined with the earlier part where we showed that the formula works for the first term is our proof by mathematical induction.
\n" ); document.write( "Mathematical induction is like setting up dominoes to fall down. The hardest part is setting up the dominoes so that if one falls then it will knock down the next one. (This is the changing the formula for n into the formula for (n+1) part.) Then the last detail is to knock down the first domino. After all, no dominoes fall at all unless it gets started! (This is the \"prove it for the first term part.) \n" ); document.write( "