how to find triangular number and square number
full explanation
Answers
Step-by-step explanation:
In each case, the two square roots involved multiply to give sk: 5 × 7 = 35, 12 × 17 = 204, and 29 × 41 = 1189. 36 − 1 = 35, 1225 − 36 = 1189, and 41616 − 1225 = 40391. In other words, the difference between two consecutive square triangular numbers is the square root of another square triangular number.
Step-by-step explanation:
Triangle numbers:
A triangular number (also known as triangle number) include objects organized in an equilateral triangle. The nth triangular number is the number of black dots in the triangular pattern with n black dots on a side and is equivalent to the total of the "n" natural numbers from "1" to "n". The arrangement of triangular numbers, beginning at the 0th triangular number, is:
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630,
The triangle numbers can be calculated by the formulas given below:
Follow the picture
Where (n+1)/2 is the Binomial coefficient. It shows the number of distinct pairs that can be chosen from (n + 1) objects, and it is said as "n plus one to one choose two".
The first equation can be represented using an image. For every triangular number Tn, think of a "half-square" pattern of objects corresponding to the triangular number, as shown in the image below. Replicating this pattern and turning it upside down to create a rectangular image doubles the number of objects, giving a rectangle with dimensions n×(n+1), which is also the number of objects in the rectangle. It clearly shows that the triangular number itself is in every case precisely half of the number of objects in such an image or Tn= (n (n+1)/2).
Square numbers:
You get a square number by multiplying a number by itself, so knowing the square numbers is a handy way to remember part of the multiplication table. Although you probably remember without help that 2 2 = 4, you may be sketchy on some of the higher numbers, such as 7 7 = 49. Knowing the square numbers gives you another way to etch that multiplication table forever into your brain.
The following figure shows the first few square numbers: 1, 4, 9, 16, and 25.
Follow the picture
From here, you can determine more square numbers:
36 49 64 . . . .
Visual aids can help you find square numbers. The tastiest visual aids you’ll ever find are those little square cheese-flavored crackers. (You probably have a box sitting somewhere in the pantry. If not, saltine crackers or any other square food works just as well.) Shake a bunch out of a box and place the little squares together to make bigger squares, as shown in the above figure.
