Question

Triangular Numbers A triangular number is a term of the sequence $$ u_{1}=1 \quad u_{n+1}=u_{n}+(n+1) $$ List the first seven triangular numbers.

Short answer

1, 3, 6, 10, 15, 21, 28

Step-by-step solution

Understand the Sequence

Triangular numbers form a sequence where each term is the sum of the natural numbers up to that term. Mathematically, the sequence is given by the recurrence relation: Initial Term: u_{1}=1 Recursive Relation: u_{n+1}=u_{n}+(n+1)

Calculate the First Term

The first triangular number is given: u_{1} = 1

Calculate the Second Term

Using the recurrence relation for the second term: u_{2} = u_{1} + 2 u_{2} = 1 + 2 = 3

Calculate the Third Term

Using the recurrence relation for the third term: u_{3} = u_{2} + 3 u_{3} = 3 + 3 = 6

Calculate the Fourth Term

Using the recurrence relation for the fourth term: u_{4} = u_{3} + 4 u_{4} = 6 + 4 = 10

Calculate the Fifth Term

Using the recurrence relation for the fifth term: u_{5} = u_{4} + 5 u_{5} = 10 + 5 = 15

Calculate the Sixth Term

Using the recurrence relation for the sixth term: u_{6} = u_{5} + 6 u_{6} = 15 + 6 = 21

Calculate the Seventh Term

Using the recurrence relation for the seventh term: u_{7} = u_{6} + 7 u_{7} = 21 + 7 = 28

Key concepts

sequence

A sequence is essentially a list of numbers in a specific order. In this context, a sequence refers to the series of triangular numbers. The key idea is that every term in the sequence has a distinct position, often denoted by an index like n. The triangular number sequence starts from an initial term, and each subsequent term is derived based on a set rule or pattern. For example, the sequence of triangular numbers begins with 1, then 3, then 6, and continues to grow.

This sequence can be understood as:

• First term: 1
• Second term: 3
• Third term: 6
• Fourth term: 10
• Fifth term: 15
• Sixth term: 21
• Seventh term: 28

Each of these numbers represents a triangular number. To get the next triangular number in the sequence, just add the next natural number to the most recent term. This pattern is crucial in what makes triangular numbers unique and easy to calculate once understood.

recurrence relation

A recurrence relation is a mathematical way to define the terms of a sequence based on the preceding terms. It provides a way to compute new terms using previous ones.

In the case of triangular numbers, the recurrence relation is:
\[ u_{n+1} = u_{n} + (n+1) \]
This means that each new term in the sequence is found by taking the previous term and adding the next natural number. The initial term is\( u_{1}=1 \). Consequently, the sequence continues:

• \(u_{2} = u_{1} + 2 = 1 + 2 = 3\)


• \(u_{3} = u_{2} + 3 = 3 + 3 = 6\)


• \(u_{4} = u_{3} + 4 = 6 + 4 = 10\)


• \(u_{5} = u_{4} + 5 = 10 + 5 = 15\)


• \(u_{6} = u_{5} + 6 = 15 + 6 = 21\)


• \(u_{7} = u_{6} + 7 = 21 + 7 = 28\)

Using the recurrence relation and the initial term allows us to generate the first seven triangular numbers.

natural numbers

Natural numbers are the set of positive integers starting from 1 and increasing by 1 each time, such as 1, 2, 3, and so forth. These numbers are crucial as they form the basis for triangular numbers.

Triangular numbers are created by summing consecutive natural numbers. Let's see how this works:

• The first triangular number is just the first natural number: 1


• The second triangular number is the sum of the first two natural numbers: 1 + 2 = 3


• The third triangular number is the sum of the first three natural numbers: 1 + 2 + 3 = 6


• Continuing this way, the fourth triangular number is: 1 + 2 + 3 + 4 = 10


• The fifth triangular number is: 1 + 2 + 3 + 4 + 5 = 15


• The sixth triangular number is: 1 + 2 + 3 + 4 + 5 + 6 = 21


• Finally, the seventh triangular number is: 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28

Understanding that triangular numbers are built from the continuous sum of natural numbers simplifies their calculation. In essence, knowing natural numbers allows us to understand and compute these fascinating triangular numbers.