Difference between revisions of "Baka to Tesuto to Syokanju:Volume7 The Fourth Question"
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| − | Use mathematical induction <sup>[http://www.baka-tsuki.net/project/index.php?title=Baka_to_Tesuto_to_Syokanju:Volume7_Translator%27s_Notes]</sup> |
+ | Use mathematical induction <sup>[http://www.baka-tsuki.net/project/index.php?title=Baka_to_Tesuto_to_Syokanju:Volume7_Translator%27s_Notes]</sup> to prove the following: |
| − | to prove the following: |
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1 + 3 + 5 + … + (2n-1)=n<sup>2</sup> – (1) |
1 + 3 + 5 + … + (2n-1)=n<sup>2</sup> – (1) |
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2. Assuming that n=k is valid, |
2. Assuming that n=k is valid, |
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| − | + | :1 + 3 + 5 + … + (2k-1)=k<sup>2</sup> - (2) |
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| − | + | When n=k+1, the left hand side of (1) is written as |
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| − | + | :1 + 3 + 5 + … + (2k-1)+(2k+1) |
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| − | + | :=k<sup>2</sup> + (2k+1) (Obtained from equation (2)) |
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| − | + | :=(k+1)<sup>2</sup> |
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In other words, |
In other words, |
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| − | + | :1 + 3 + 5 + … + (2k-1)+(2k+1) |
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| − | + | :=(k+1)<sup>2</sup> |
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(1) is valid when n=k+1. |
(1) is valid when n=k+1. |
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Revision as of 12:50, 5 September 2011
Use mathematical induction [1] to prove the following:
1 + 3 + 5 + … + (2n-1)=n2 – (1)
- When n is a natural number.
Himeji Mizuki's answer
1. When n=1, equation (1) is calculated as,
- Left Hand Side = 1
- Right Hand Side = 1
2. Assuming that n=k is valid,
- 1 + 3 + 5 + … + (2k-1)=k2 - (2)
When n=k+1, the left hand side of (1) is written as
- 1 + 3 + 5 + … + (2k-1)+(2k+1)
- =k2 + (2k+1) (Obtained from equation (2))
- =(k+1)2
In other words,
- 1 + 3 + 5 + … + (2k-1)+(2k+1)
- =(k+1)2
(1) is valid when n=k+1.
From both equations (1) and (2), equation (1) is valid for all values on n.
Teacher's comment
That's correct. In mathematical induction, you need to prove that n=1 is valid. Also, when assuming that n=k is valid, you have to deduce that n=k+1 is valid. In this question, the key is that n is a natural number. On a side note, people often forget to put that n=1, so do take note when answering.
Tsuchiya Kouta's answer
"I'll prove here that (1) is correct - Tsuchiya Kouta"
Teacher's comment
Even if you try to sneak through by writing it in a style of a thesis, it's usless. The question mentioned that you have to prove it through 'mathematical induction', so while assuming that n=k, please write the equation that n=k+1 is valid.
Yoshii Akihisa's answer
"My judgment can hold."
Teacher's comment
"Please use an assumption."
