# Induction proof of harmonic triangle formula

• timothychoi
In summary, the conversation discusses the formula for the Leibniz harmonic triangle and the use of induction to prove its assertion. It is mentioned that there may be an error in the formula on Wikipedia and a link to the correct formula on Mathworld is provided.
timothychoi
Hello. In the Wiki entry for "Leibniz harmonic triangle" there one
can find a formula

L(r, c) = 1/(c * binom(r - 1, c - 1))

where L(r, c) is the entry in the harmonic triangle and the
binom() is the usual binomial notation (or the entry from
Pascal triangle). I tried to prove the assertion using indunction
on n, then I was not able to finish. Can someone prove
the assertion using induction on n? Only the last step, that
is assuming that

L(k, c) = 1/(c * binom(k - 1, c - 1))

and showing that

L(k + 1, c) = 1/(c * binom((k + 1) - 1, c - 1)),

will be sufficient. Thank you.

timothychoi said:
Hello. In the Wiki entry for "Leibniz harmonic triangle" there one
can find a formula

L(r, c) = 1/(c * binom(r - 1, c - 1))

where L(r, c) is the entry in the harmonic triangle and the
binom() is the usual binomial notation (or the entry from
Pascal triangle). I tried to prove the assertion using indunction
on n, then I was not able to finish. Can someone prove
the assertion using induction on n? Only the last step, that
is assuming that

L(k, c) = 1/(c * binom(k - 1, c - 1))

and showing that

L(k + 1, c) = 1/(c * binom((k + 1) - 1, c - 1)),

will be sufficient. Thank you.

I think that should be

L(r, c) = 1/(r * binom(r - 1, c - 1))

-evidently an error on Wikipedia. Compare the entry on Mathworld:

http://mathworld.wolfram.com/LeibnizHarmonicTriangle.html

## 1. What is the harmonic triangle formula?

The harmonic triangle formula is a mathematical formula used to find the sum of the reciprocals of the numbers in a given row of Pascal's triangle. It is written as S(n,k) = (n+1)/(k+1), where n is the row number and k is the position in the row.

## 2. How is induction used to prove the harmonic triangle formula?

Induction is a mathematical technique used to prove a statement for all natural numbers. In the case of the harmonic triangle formula, induction is used to show that the formula holds true for every row of Pascal's triangle. This is done by first proving the formula for the base case (usually row 1), and then assuming it is true for a particular row and proving it for the next row. By repeating this process for all rows, the formula is proven for all natural numbers.

## 3. Why is the harmonic triangle formula important?

The harmonic triangle formula has many applications in mathematics, particularly in the field of combinatorics. It can be used to solve problems involving combinations and probabilities, and it also has connections to other mathematical concepts such as binomial coefficients and the Fibonacci sequence. Additionally, understanding and being able to use the harmonic triangle formula can help in developing critical thinking and problem-solving skills.

## 4. Can the harmonic triangle formula be extended to other triangles?

Yes, the harmonic triangle formula can be extended to other types of triangles, such as the Catalan triangle and the Stirling triangle. These triangles have their own respective formulas for finding the sum of reciprocals, which can also be proven using induction.

## 5. Are there any real-world applications of the harmonic triangle formula?

Yes, the harmonic triangle formula has practical applications in fields such as computer science, physics, and engineering. It can be used to solve problems involving combinations and probabilities, as well as in the analysis of data and signal processing. Additionally, the harmonic triangle formula can be used to model and understand various natural phenomena, such as sound waves and population growth.

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