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Chapter 1: Problem 2

If \(t_{n}\) denotes the \(n\) th triangular number, prove that in terms of the binomial coefficients, $$ t_{n}=\left(\begin{array}{c} n+1 \\ 2 \end{array}\right) \quad n \geq 1 $$

Short Answer

Expert verified
The nth triangular number equals \( \binom{n+1}{2} \).

Step by step solution

01

Understanding Triangular Numbers

Triangular numbers are a sequence of numbers where each number represents a triangle with dots. The general formula for the nth triangular number is given by \( t_n = \frac{n(n+1)}{2} \).
02

Understanding Binomial Coefficient

The binomial coefficient \( \binom{n+1}{2} \) represents the number of ways to choose 2 elements from \( n+1 \) elements and is defined as \( \binom{n+1}{2} = \frac{(n+1)!}{2!(n-1)!} \).
03

Simplification of the Binomial Coefficient

The expression for the binomial coefficient simplifies to \( \binom{n+1}{2} = \frac{(n+1)n}{2} \) because \( 2! = 2 \) and \( (n+1)! = (n+1) \cdot n \cdot (n-1)! \).
04

Equivalence of Formulas

By comparing the expression \( \frac{n(n+1)}{2} \) for triangular numbers with the expression \( \frac{(n+1)n}{2} \) for the binomial coefficient, we can see that they are equal, thus proving \( t_n = \binom{n+1}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Coefficients
Binomial coefficients are fundamental in mathematics, especially in the field of combinatorics. They describe the number of ways to choose a subset of elements from a larger set. For example, when you see the symbol \( \binom{n}{k} \), it represents the number of ways to pick \( k \) elements from a set of \( n \) elements without considering the order of selection. This can be calculated using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
Understanding binomial coefficients is crucial because they appear frequently in algebra and are also essential in understanding the binomial theorem. For instance, the coefficients in the expansion of \((a+b)^n\) are precisely the binomial coefficients.
Moreover, binomial coefficients help us connect various mathematical concepts. In this exercise, they serve to equate a well-known sequence, the triangular numbers, allowing us to prove \( t_n = \binom{n+1}{2} \). They ensure that the conceptual link between the arrangement of objects and algebraic expressions remains intact and insightful.
Combinatorics
Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. It plays a vital role in solving problems involving probability, logic, and algebra.
One of the main aspects of combinatorics is to determine the number of ways of arranging certain objects or choosing items from a group. For instance, selecting lottery numbers, distributing objects among people, and organizing events are all combinatorial in nature.
In the context of triangular numbers, combinatorics helps us determine the number of ways two items can be chosen from \( n+1 \) items, which is exactly \( \binom{n+1}{2} \). This relates directly to understanding why the formula for the nth triangular number, \( t_n = \frac{n(n+1)}{2} \), can equivalently be expressed using a binomial coefficient.
Mathematical Proof
A mathematical proof is a compelling argument that verifies the truth of a mathematical statement. It uses logical reasoning, definitions, previously proven statements, and axioms to justify the validity of the statement.
In proving that triangular numbers can be expressed through binomial coefficients, a proof demonstrates that the formulas \( t_n = \frac{n(n+1)}{2} \) and \( \binom{n+1}{2} = \frac{(n+1)n}{2} \) are indeed equivalent.
  • The proof starts by using the known formula for triangular numbers \( t_n = \frac{n(n+1)}{2} \).
  • We then analyze the binomial coefficient \( \binom{n+1}{2} \), which simplifies to \( \frac{(n+1)n}{2} \).
  • Comparing these expressions confirms their equality, thereby proving that \( t_n = \binom{n+1}{2} \).
Such proofs are not only about solving a problem but also about enhancing the comprehension of underlying mathematical structures, reinforcing the importance of trusted methods in exploring and understanding mathematics.

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Most popular questions from this chapter

Prove that \(n !>n^{2}\) for every integer \(n \geq 4\), whereas \(n !>n^{3}\) for every integer \(n \geq 6\).

Prove that the cube of any integer can be written as the difference of two squares. [Hint: Notice that $$ \left.n^{3}=\left(1^{3}+2^{3}+\cdots+n^{3}\right)-\left(1^{3}+2^{3}+\cdots+(n-1)^{3}\right) .\right] $$

The Catalan mumbers, defined by $$ C_{n}=\frac{1}{n+1}\left(\begin{array}{c} 2 n \\ n \end{array}\right)=\frac{(2 n) !}{n !(n+1) !} \quad n=0,1,2, \ldots $$ form the sequence \(1,1,2,5,14,42,132,429,1430,4862, \ldots .\) They first appeared in 1838 when Eug猫ne Catalan ( \(1814-1894\) ) showed that there are \(C_{n}\) ways of parenthesizing a nonassociative product of \(n+1\) factors. [For instance, when \(n=3\) there are five ways: \(((a b) c) d,(a(b c)) d, a((b c) d), a(b(c d)),(a b)(a c) \cdot]\) For \(n \geq 1\), prove that \(C_{n}\) can be given inductively by $$ C_{n}=\frac{2(2 n-1)}{n+1} C_{n-1} $$

Establish the formulas below by mathematical induction: (a) \(1+2+3+\cdots+n=\frac{n(n+1)}{2}\) for all \(n \geq 1\). (b) \(1+3+5+\cdots+(2 n-1)=n^{2}\) for all \(n \geq 1\). (c) \(1 \cdot 2+2 \cdot 3+3 \cdot 4+\cdots+n(n+1)=\frac{n(n+1)(n+2)}{3}\) for all \(n \geq 1\) (d) \(1^{2}+3^{2}+5^{2}+\cdots+(2 n-1)^{2}=\frac{n(2 n-1)(2 n+1)}{3}\) for all \(n \geq 1\). (e) \(1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\left[\frac{n(n+1)}{2}\right]^{2}\) for all \(n \geq 1\).

For all \(n \geq 1\), prove the following by mathematical induction: (a) \(\frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\cdots+\frac{1}{n^{2}} \leq 2-\frac{1}{n}\). (b) \(\frac{1}{2}+\frac{2}{2^{2}}+\frac{3}{2^{3}}+\cdots+\frac{n}{2^{n}}=2-\frac{n+2}{2^{n}}\).
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