Can someone please explain to me Mathematical Induction as I'm just getting confused by it. I never really got it the first time and are again having problems with it. I have trouble with the third step, proving true for k + 1 usually. Thanks.
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Mathematical Induction (1 Viewer)
- Thread starter henry08
- Start date
The third step is usually the hardest and you usually have to use the second step in it. I dont know what you want explained. You just have to prove that it works for K + 1 and then as it works for 1 it will work for 2 and it will work for 3 etc.
I usually break it up into 4 steps.
Remember, Tn is on the LEFT side of the EQUAL sign.. and Sn is to the right of the equal sign.
This would be how I'd do an induction 'for all positive integers n'
Step 1
Show true for n = 1
Sub in n=1 into both Tn and Sn, they should be equal
Hence true for n = 1
Step 2
Assume true for n = k
Just sub k into both Tn and Sn to find Tk and Sk
Step 3
Show true for n = k + 1
First, sub (k+1) into Sn to find S(k+1)
Then, use this: S(k+1) = Sk + T(k+1)
Sub in T(k+1) into Tn, to find T(k+1)
Then use your results and sub into formula S(k+1) = Sk + T(k+1)
Keep simplifying and manipulating the equation (only the RHS) until u get to the result you wrote for S(k+1)
Then you write: Hence, true for n=k+1
Step 4
Since true for n=1, n=k+1, then true for n=2, n=3 and all positive integers n.
Remember, Tn is on the LEFT side of the EQUAL sign.. and Sn is to the right of the equal sign.
This would be how I'd do an induction 'for all positive integers n'
Step 1
Show true for n = 1
Sub in n=1 into both Tn and Sn, they should be equal
Hence true for n = 1
Step 2
Assume true for n = k
Just sub k into both Tn and Sn to find Tk and Sk
Step 3
Show true for n = k + 1
First, sub (k+1) into Sn to find S(k+1)
Then, use this: S(k+1) = Sk + T(k+1)
Sub in T(k+1) into Tn, to find T(k+1)
Then use your results and sub into formula S(k+1) = Sk + T(k+1)
Keep simplifying and manipulating the equation (only the RHS) until u get to the result you wrote for S(k+1)
Then you write: Hence, true for n=k+1
Step 4
Since true for n=1, n=k+1, then true for n=2, n=3 and all positive integers n.
<table id="post3770907" class="tborder" align="center" border="0" cellpadding="3" cellspacing="0" width="100%"><tbody><tr valign="top"><td class="alt1" id="td_post_3770907">ok, i get the topic but i always have trouble writing out the conclusion, step 4...
therefore n=k is true.. something or other?
what do you write, generically?
for:
- divisible
- multiple
- inequality
- summation?
cheers.
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therefore n=k is true.. something or other?
what do you write, generically?
for:
- divisible
- multiple
- inequality
- summation?
cheers.
<!-- / message --> </td> </tr> <tr> <td class="alt2">
</td> <td class="alt1" align="right"> <!-- controls -->
</td></tr></tbody></table>
davidbarnes
Trainee Mȯderatȯr
Thanks for the help. I've got a pretty good grasp on this now.