cancelling the terms on the Left Hand side, we get, n + n n n n ... + n n n n n . . Sum of cubes of first n natural numbers To find n , let us consider the identity Where x = , , ,.....n – , n x = , x = , x = , − , n , ( Adding all these equations and cancelling the terms on the Left Hand side, we get, ( n + ) – = n n n n n n n n n = n n Ideal Friendship Consider the numbers and .
Sum of the divisors of (excluding ) = + + + + + + + + + + = Sum of the divisors of (excluding ) = + + + + = . Thus, sum of divisors of one number excluding itself is the other. Such pair of numbers is called Amicable Numbers or Friendly Numbers. and are least pair of Amicable Numbers.
They were discovered by Pythagoras. We now know more than million amicable pair of Numbers. Numbers and Sequences Activity Take a triangle like this Make another triangle like this. Join the second triangle with the first to get Thus, two copies of provide a rectangle of size ´ .
We can write in numbers, what we did with pictures. Let us write, ( Therefore, In a similar, fashion, try to find