This is the sequence 1,3,6,10,15 the pattern is addin 1 more than last time but what is the name for this pattern
These are called the triangular numbers
The sequence is
1
3=1+2
6=1+2+3
10=1+2+3+4
15=1+2+3+4+5
You can also observe this pattern
x
_________
x
xx
__________
x
xx
xxx
__________
x
xx
xxx
xxxx
to see why they're called triangular numbers. I think the Pythagoreans (around 700 B.C.E.) were the ones who gave them this name. I do know the Pythagoreans tried to assign numbers to many different objects like this.
Triangular Numbers
The number of dots, circles, spheres, etc., that can be arranged in an equilateral or right triangular pattern is called a triangular number. The 10 bowling pins form a triangular number as do the 15 balls racked up on a pool table. Upon further inspection, it becomes immediately clear that the triangular numbers, T1, T2, T3, T4, etc., are simply the sum of the consecutive integers 1234.....n or Tn = n(n + 1)/2, namely, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66,78, 91, etc.
Triangular numbers are the sum of the balls in the triangle as defined by Tn = n(n + 1)/2.
Order.n...1........2...............3.....................4.............................5....................6.....7.....8.....9
.............O.......O...............O....................O............................O
....................O...O.........O...O...............O...O.......................O...O
..................................O...O...O.........O....O....O...............O....O....O
......................................................O....O...O....O.........O.....O...O....O
.................................................................................O....O....O....O....O
Total......1........3.................6....................10..........................15..................21...28...36...45...etc.
The sum of a series of triangular numbers from 1 through Tn is given by S = (n^3 + 3n^2 + 2n)/6.
After staring at several triangular and square polygonal number arrangements, one can quickly see that the 1st and 2nd triangular numbers actually form the 2nd square number 4. Similarly, the 2nd and 3rd triangulars numbers form the 3rd square number 9, and so on. By inspection, one can see that the nth square number, Sn, is equal to Tn + T(n  1) = n^2. This can best be visualized from the following:
.........Tn  1...3...6...10...15...21...28...36...45...55...66...78...91
.........T(n  1)........1...3....6....10...15...21...28...36...45...55...66...78
.........Sn.........1...4...9....16...25...36...49...64...81..100.121.144.169
A number cannot be triangular if its digital root is 2, 4, 5, 7 or 8.
Some interesting characteristice of Triangular numbers:
The numbers 1 and 36 are both square and triangular. Some other triangular squares are 1225, 41,616, 1,413,721, 48,024,900 and 1,631,432,881. Triangular squares can be derived from the series 0, 1, 6, 35, 204, 1189............Un where Un = 6U(n  1)  U(n  2) where each term is six times the previous term, diminished by the one before that. The squares of these numbers are simultaneously square and triangular.
The difference between the squares of two consecutive rank triangular numbers is equal to the cube of the larger numbers rank.
Thus, (Tn)^2  (T(n  1))^2 = n^3. For example, T6^2  T5^2 = 441  225 = 216 = 6^3.
The summation of varying sets of consecutive triangular numbers offers some strange results.
T1 + T2 + T3 = 1 + 3 + 6 = 10 = T4.
T5 + T6 + T7 + T8 = 15 + 21 + 28 + 36 = 100 = 45 + 55 = T9 + T10.
The pattern continues with the next 5 Tn's summing to the next 3 Tn's followed by the next6 Tn's summing to the next 4 Tn's, etc.
The sum of the first "n" cubes is equal to the nth triangular number. For instance:
n............1.....2.....3.....4.......5
Tn..........1.....3.....6....10.....15
n^3.........1 + 8 + 27 + 64 + 100 = 225 = 15^2
Every number can be expressed by the sum of three or less triangular numbers, not necessarily different.
1 = 1, 2 = 1 + 1, 3 = 3, 4 = 3 + 1, 5 = 3 + 1 + 1, 6 = 6, 7 = 6 + 1, 8 = 6 + 1 + 1, 9 = 6 + 3, 10 = 10, etc.
Alternate ways of finding triangular squares.
From Tn = n(n + 1)/2 and Sn = m^2, we get m^2 = n(n + 1)/2 or 4n^2 + 4n = 8m^2.
Adding one to both sides, we obtain 4n^2 + 4n + 1 = 8m^2 + 1.
Factoring, we find (2n + 1)^2 = 8m^2 + 1.
If we allow (2n + 1) to equal "x" and "y" to equal 2m, we come upon x^2  2y^2 = 1, the famous Pell Equation.
We now know that the positive integer solutions to the Pell equation, x^2  2y^2 = +1 lead to triangular squares. But how?
Without getting into the theoretical aspect of the subject, sufficeth to say that the Pell equaion is closely connected with early methods of approximating the square root of a number. The solutions to Pell's equation, i.e., (x,y), often written as (x/y) are approximations of the square root of D in x^2  2y^2 = +1. Numerous methods have evolved over the centuries for estimating the square root of a number.
Diophantus' method leads to the minimum solutions to x^2  Dy^2 = +1, D a non square, by setting x = my + 1 which leads to y = 2m/(D  m^2).
From values of m = 1.......n, many rational solutions evolve.
Eventually, an integer solution will be reached.
For instance, the smallest solution to x^2  2y^2 = +1 derives from m = 1 resulting in x = 3 and y = 2 or sqrt(2) ~= 3/2..
Newton's method leads to the minimum solution sqrt(D) = sqrt(a^2 + r) = (a + D/a)/2 ("a" = the nearest square) = (3/2).
Heron/Archimedes/El Hassar/Aryabhatta obtained the minimum solution sqrt(D) = sqrt(a^2 +r) = a +r/2a = (x/y) = (3/2).
Other methods exist that produce values of x/y but end up being solutions to x^2  Dy^2 = +/C.
Having the minimal solutions of x1 and y1 for x^2  Dy^2 = +1, others are derivable from the following:
(x + ysqrtD) = (x1 + y1sqrtD)^n, n = 1, 2, 3, etc.
Alternitive approach
Given x = p and y = q satifying x^2  2y^2 = +1, we can write (x + sqrtD)(x  sqrtD) = 1.
x = [(p + qsqrt(2))^n + (p  qsqrt(2))^n]/2
y = [(p + qsqrt(2))^n  (p  qsqrt(2))^n]/(2sqrt(2))
Having the minimum solution of x = 3 and y = 2, the next few solutions derive from n = 2 and 3 where x = 17, y = 12, x = 99 and y = 70 respectively.
Alternative approach
Subsequent solutions can also be obtained by means of the following:
x^2  2y^2 = +1 can be rewritten as x^2  2y^2 = (x + yqrt(2)(x  ysqrt(2)) = +1.
Using the minimum solution of x = 3 and y = 2, we can now write
.................(3 + sqrt(2))^2(3  sqrt(2))^2 = 1^2 = 1
.................(17 + 12sqrt(2))(17  12sqrt(2)) = 1
.................289  2(144) = 17^2  2(12)^2 = 1 the next smallest solution.
The next smallest solution is derivable from
.................(3 + sqrt(2))^3(3  sqrt(2))^3 = 1^2 = 1 which works out to
.................(99 + 70sqrt(2))(99  70sqrt(2)) = 1 or
.................99^2  2(70)^2 = 1.
Similarly, (3 + sqrt(2))^4(3  sqrt(2))^4 = 1^2 = 1 leads to
.................(577 + 408sqrt(2))(577  408sqrt(2)) = 1 and
.................577^2  2(408)^2 = 1.
Regardless of the method, we ultimately end up with the starting list of triangular squares.
..x........y........n.......m........Tn = Sm^2
..3........2........1........1..............1
.17......12.......8........6..............36
.99......70......49......35...........1225
577....408....288.....204.........41,616 etc.
nb [
 👍
 👎
 👁
 ℹ️
 🚩
1 answer

It appears that you have answered your own question. Let us know if you need further assistance.
 👍
 👎
 ℹ️
 🚩
Answer this Question
Related Questions

Art
1. Which of the following design principles is evident on this dish? A. texture and harmony B. asymmetrical balance and pattern C. radial balance and pattern D. symmetrical balance and harmony My answer: C 2. The usage of white lines retreating towards the

Ed. Tech
Which definition below best describes the definition of theme within poetry? A. It is the pattern of rhyming lines within a poem. B. It is the underlying message that a poem conveys. C. It is the pattern of stressed and unstressed syllables in a poem. D.

Maths
So there's a sequence: 0, 1/2, 3/4, 7/8, 15/16,... and I have to find the pattern and the next two numbers. I don't understand how I should go about doing this. I don't believe it uses multiplication or division as it starts with 0. Help?

Mathplease check
1. Write a rule for the sequence. 8, 1, 10, 19... A. Start with 8 and add 9 repeatedly B. star with 9 and add 8 repeatedly C. start with 8 and add 9 repeatedly D. start with 8 and subtract 9 repeatedly 3. What is the 7th term in the following

algebra
The second term in a geometric sequence is 20. The fourth term in the same sequence is 45/4, or 11.25. What is the common ratio in the sequence? Thanks for reading :)

probability
consider a sequence of independent tosses of a biased coin at times k=0,1,2,…,n. On each toss, the probability of Heads is p, and the probability of Tails is 1−p. A reward of one unit is given at time k, for k∈{1,2,…,n}, if the toss at time k

math
1,5,11,19 (a) calculate next two patterns (b) calculate the nth term of the pattern (c)which term of the pattern is equal to 2549. thanks a lot

art
The math tessellations: unit 4 1. The architectural design was created by: 1. using an organic pattern in the form of tessellation. 2. using a geometric pattern in the form a tessellation. 3. using no definite pattern and placing shapes randomly. 4. using

Can someone help me?!
The 1st, 5th and 13th terms of an arithmetic sequence are the first three terms of a geometric sequence with a common ratio 2. If the 21st term of the arithmetic sequence is 72, calculate the sum of the first 10 terms of the geometric sequence.

math
The 1st,5th,13th term of an arithmetic sequence are the first 3 terms of geometric sequence with a common ratio of 2. If the 21st term of the arithmetic sequence is 72, calculate the sum of the first 10 terms of the geometric sequence.

math
Patterns are everywhere. Some of them are geometrical and some of them are numerical. What is one way in which patterns can be used? Give a realworld example of a pattern and identify the terms and the sequence.

Sequences
Like the Fibonacci sequence, a certain sequence satisfies the recurrence relation an=an−1+an−2. Unlike the Fibonacci sequence, however, the first two terms are a1=4 and a2=1. Find a32.

algebra
Describe the pattern in each sequence and determine the next term of the sequence. 11, 17, 23, 29, … The pattern in each sequence is that after each sequence, the number is added by 6. The next term of the sequence would be 35. Is this right?

mathematics
You generated a pattern using the rule “multiplied by 2 then subtract by 1”. You started from the number 2. What are the first five terms of your pattern?

math HELP PLEASE
Given the pattern rule, write the first 5 terms for each sequence: a. Start with 0, add 5 b. Start with 2, multiply by 6 c. Start with 100, divide by 2 and add 10 d. Start with 1, multiply by 3, subtract 1

Bio
How is information for a specific protein carried on the DNA molecule? A. As a sequence of nucleotides B. In the double helix shape of the condensed chromosome C.In the ratio of adenines to thymines D.As a pattern of phosphates and sugars A C D are all

Math
i need help with problem d Graph the arithmetic sequence an = –4, –1, 2, 5, 8, … a. What is the 8th term in the sequence? a_ n = a_1 + (n – 1) d. a_8 = 4 + 7(3) = 17 b. What is the recursive formula for this sequence? A_n+1 = a_n + 3 c. What is

Algebra 1 Honors
Write an equation for the nth term of the arithmetic sequence. Then graph the first five terms in the Sequence. 3,8,13,18 I got 5n8 as my formula. Then plugged 5 in for n and got 17 for the 5th number in the sequence. Is this correct?

Math Alg. 2
Write a recursive rule for the sequence: 1,2,12,56,272... and 2,5,11,26,59... and 3,2,5,3,2... I can't find the pattern in these and am unsure on how to write the rule. Thanks

algebra
Identify the pattern and find the next number in the pattern48,24,12,6 answers are 3, 0, 2, 3
Still need help?
You can ask a new question or browse existing questions.