on the square roots of triangular numbers It is well known that the balancing numbers are the square roots of the triangular numbers and are the solutions of the Diophantine equation 1 2 n 1 n
100 MAY ON THE SQUARE ROOTS OF TRIANGULAR NUMBERS c B2n B2n B2n v d B 2n l B n Bn l Bn l Proof From 9 it follows that Bn l Write Nk for the kth square triangular number and write sk and tk for the sides of the corresponding square and triangle so that Define the triangular root of a triangular number N n n 1 2 to be n From this definition and the quadratic formula Therefore N is triangular n is an integer if and only if 8N 1 is square Consequently a square
on the square roots of triangular numbers
on the square roots of triangular numbers
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My Math Resources Squares Square Roots Poster
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A24a Recognising And Using Square Cube And Triangular Numbers
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By analogy with the square root of x one can define the positive triangular root of x as the number n such that Tn x which follows immediately from the quadratic formula So an integer x is triangular if and only if 8x 1 is a square Equivalently if the positive triangular root n of x is an integer then x is the nth triangular number In the joint paper On the square roots of triangular numbers published in The Fibonacci Quarterly in 1999 Behera and Panda introduced balancing numbers and studied many important
Triangular numbers are numbers that can be represented as a triangle The numbers form a sequence known as the triangular numbers The first triangular number T 1 1 T 1 1 The second triangular Formula The formula to find the n th triangular number is T n i 1 n i n n 1 2 Derivation As we know the sum of the first n natural numbers can be
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Proof S ince x is a balancing num ber 8x2 1 is a perfect square and 8x2 8x2 l 4 x 2 8 x 2 1 is a triangular num ber w hich is also a perfect square therefore its An arrangement of triangular numbers starting with the 0th triangular number is as follows Numbers 0 through 1 3 6 10 15 21 28 36 45 55 66 78 91
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on the square roots of triangular numbers - In the joint paper On the square roots of triangular numbers published in The Fibonacci Quarterly in 1999 Behera and Panda introduced balancing numbers and studied many important