Harmonic Numbers In Harmony

Algebra Level 1

True or False?

1729 H 1728 − 1728 = 1729 H 1729 − 1729. {1729}H_{{1728}}-{1728}={1729}H_{{1729}}-{1729}.

Notation : H n H_n denote the n th n^\text{th} harmonic number , H n = 1 + 1 2 + 1 3 + ⋯ + 1 n H_n = 1 + \frac12 + \frac13 + \cdots + \frac1n .

True False

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Nihar Mahajan by Nihar Mahajan , 20, India

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2 solutions

Nihar Mahajan
Feb 19, 2016

1729 H 1728 − 1728 = 1729 H 1729 − 1729 ⇒ 1729 H 1728 − 1728 + 1729 = 1729 H 1729 ⇒ 1729 H 1728 + 1 = 1729 H 1729 1729H_{1728}-1728=1729H_{1729}-1729 \\ \Rightarrow 1729H_{1728} -1728+1729=1729H_{1729} \\ \Rightarrow 1729H_{1728} +1=1729H_{1729}

Dividing both sides by 1729 1729 we have:

H 1728 + 1 1729 = H 1729 H_{1728}+\dfrac{1}{1729}=H_{1729}

which is true since H n − 1 + 1 n = H n H_{n-1} +\dfrac{1}{n}=H_{n} .

29 Helpful 2 Interesting 2 Brilliant 1 Confused

Moderator note:

Simple standard approach.

Alternatively, we could do this:

H 1729 − H 1728 = 1 / 1729 1729 ( H 1729 − H 1728 ) = 1 1729 ( H 1729 − H 1728 ) = 1729 − 1728 1729 H 1729 − 1729 H 1728 = 1729 − 1728 1729 H 1729 − 1729 = 1729 H 1728 − 1728 \begin{aligned} H_{1729} - H_{1728} &=& 1/1729 \\ 1729(H_{1729} - H_{1728} )&=& 1 \\ 1729(H_{1729} - H_{1728} )&=& 1729 - 1728 \\ 1729H_{1729} -1729 H_{1728} &=& 1729 - 1728 \\ 1729H_{1729} - 1729& =& 1729H_{1728} - 1728 \end{aligned}

Pi Han Goh - 5 years, 3 months ago

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Yes , nice way. You like Harmonic numbers it seems :3

Nihar Mahajan - 5 years, 3 months ago

The only problem is the proof itself. You shouldn't be aiming to get to a point where something is true. Just swap the order of statements around like Pi's proof and it'll be valid.

Josh Banister - 4 years, 11 months ago

when I mark "true" it says "incorrect" and yet the proof says it is true. What's up with that?

Samuel Prieto - 10 months ago

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@Brilliant Mathematics , need help.

Pi Han Goh - 10 months ago

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@Samuel Prieto , the system shows that you didn't submit an answer.

Brilliant Mathematics Staff - 9 months, 4 weeks ago

Doesn't 1729 H 1729 = 1 ? 1729H_{1729} = 1?

Andrew Tawfeek - 5 years, 3 months ago

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H 1729 = 1 + 1 2 + 1 3 + ⋯ 1 1729 H_{1729} = 1+ \dfrac {1}{2}+ \dfrac {1}{3}+ \cdots \dfrac {1}{1729}

Mehul Arora - 5 years, 3 months ago

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Ohh, thank you!

Andrew Tawfeek - 5 years, 3 months ago

H n {H}_{n} is the sum of all unit fractions from 1 to n

Hamza A - 5 years, 3 months ago

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Thank you! I accidentally thought it was the nth term in the harmonic series :)

Andrew Tawfeek - 5 years, 3 months ago

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@Andrew Tawfeek – happy to help :)

Hamza A - 5 years, 3 months ago
Chew-Seong Cheong
Oct 15, 2016

1729 H 1728 − 1728 = 1729 H 1728 + 1 − 1729 = 1729 ( 1 + 1 2 + 1 3 + . . . + 1 1728 ) + 1 − 1729 = 1729 ( 1 + 1 2 + 1 3 + . . . + 1 1728 + 1 1729 ) − 1729 = 1729 H 1729 − 1729 \begin{aligned} 1729H_{1728} \color{#3D99F6}{ -1728} & = 1729H_{1728} \color{#3D99F6}{+1 -1729} \\ & = 1729\left(1+\frac 12 + \frac 13 + ... + \frac 1{1728} \right)+ \color{#D61F06}{1} \color{#333333}{ - 1729} \\ & = 1729\left(1+\frac 12 + \frac 13 + ... + \frac 1{1728} + \color{#D61F06}{\frac 1{1729}} \right)- 1729 \\ & = \boxed{1729H_{1729} -1729} \end{aligned}

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