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Algebra

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 Joined: 25 Feb 2011 Post: 6
10 Jul 2011 22:54:58 IST
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Find all triangular numbers that are perfect squares.
Engineering Entrance , JEE Main , JEE Main & Advanced , Mathematics , Algebra

Find all triangular numbers that are perfect squares.

Blazing goIITian

Joined: 27 Dec 2010
Posts: 1508
11 Jul 2011 12:45:42 IST
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A triangular number or triangle number is the number of dots in an equilateral triangle uniformly filled with dots. For example, three dots can be arranged in a triangle; thus three is a triangle number. The nth triangle number is the number of dots in a triangle with n dots on a side.

the nth triangular number is given by n(n+1)/2.

the sum of two consecutive triangular numbers is a square number with the sum being the square of the difference between the two.

Tn + T(n+1) = n2

Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n..

cause [n(n+1)/2]2  =   sum of all the cubes of the individual natural numbers from one 1 to n.

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ....   this is how they go.

if a perfect number is a whole square then, n(n+1)/2  = m2

for example 1 and 36.

also all perfect numbers are triangular.

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16 Jul 2011 23:53:25 IST
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