Explanation Given information: The triangular number in terms of a sequence u 1 = 1 ; u n + 1 = u n + ( n + 1 ) . Formula used: The sum of natural numbers from 1 to n + 1 is represented by ( n + 1 ) ( n + 2 ) 2 . Proof: Let u 1 = 1 ; u n + 1 = u n + ( n + 1 ) That is, u n + 1 = u n + ( n + 1 ) = 1 + 2 + 3 + .... + n + ( n + 1 ) = n ( n + 1 ) 2 + ( n + 1 ) By simplifying, u n + 1 = ( n + 1 ) ( n + 2 ) 2 Consider the equilateral triangle. In the triangle, place the objects on the lines, starting with first object on the first line, second object on the second line, third object on the third line and so on until the last object n + 1 on the line n + 1 . Therefore, the triangular number counts objects arranged in an equilateral triangle. The n + 1 th triangular number is the number of objects on the triangular arrangement with n + 1 objects on each sides and The triangular numbers is equal to the sum of the n + 1 natural number from 1 to n + 1 that represented by ( n + 1 ) ( n + 2 ) 2 . Here, u n + 1 = ( n + 1 ) ( n + 2 ) 2 Thus, the terms of a sequence u 1 = 1 ; u n + 1 = u n + ( n + 1 ) called the triangular numbers

Precalculus Plus MyLab Math with eText -- Access Card Package (10th Edition) (Sullivan & Sullivan Precalculus Titles)

10th Edition

ISBN: 9780321978981

Author: Michael Sullivan

Publisher: PEARSON

See similar textbooks

Concept explainers

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

Videos

Question

Chapter 12.1, Problem 101AYU

To determine

To prove: The numbers 1,3,6,10,15,21,28 are called triangular.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12.1, Problem 100AYU

Chapter 12.1, Problem 102AYU

Students have asked these similar questions

On from the equation: 2 u = C₁ + C₂ Y + Czy + Cu y³ Find C₁, C₂, C3 and Cy Using these following Cases : (a) 4=0 at y=0 (b) U = U∞ at y = 8 du (c) at Y = S ду --y. ди = 0 at y = 0 буг

Tips S ps L 50. lim x2 - 4 x-2x+2 51. lim 22 - X 52. 53. x 0 Answer lim x 0 lim 2-5 X 2x2 2 x² Answer -> 54. lim T - 3x - - 25 +5 b+1 b3b+3 55. lim X x-1 x 1 Answer 56. lim x+2 x 2 x 2 57. lim x²-x-6 x-2 x²+x-2 Answer-> 23-8 58. lim 2-22-2

S 36. lim 5x+2 x-2 37. lim √√2x4 + x² x-3 Answer-> 2x3 +4 38. lim x12 √ x² + 1 √√x² + 8 39. lim x-1 2x+4 Answer 40. lim x3 2x x√x² + 7 √√2x+3

Chapter 12 Solutions

Precalculus Plus MyLab Math with eText -- Access Card Package (10th Edition) (Sullivan & Sullivan Precalculus Titles)

Ch. 12.1 - For the function f( x )= x1 x , find f( 2 ) and f(...Ch. 12.1 - True or False A function is a relation between two...Ch. 12.1 - Prob. 3AYU Ch. 12.1 - True or False The notation a 5 represents the...Ch. 12.1 - True or False If is am integer, then Ch. 12.1 - The sequence a 1 =5 , a n =3 a n1 is an example of...Ch. 12.1 - The notation a 1 + a 2 + a 3 ++ a n = k=1 n a k...Ch. 12.1 - Prob. 8AYU Ch. 12.1 - In Problems 11-16, evaluate each factorial...Ch. 12.1 - Prob. 10AYU

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

To prove: The numbers 1 , 3 , 6 , 10 , 15 , 21 , 28 are called triangular.

Solution Summary: The author explains that the triangular number is equal to the sum of the n+1 natural number from 1 to

Concept explainers

Videos

To prove: The numbers 1,3,6,10,15,21,28 are called triangular.

Want to see the full answer?

Chapter 12 Solutions