In the 14th century text Ganitasarakaumudi the following rul

In the 14th century text Ganitas¯arakaumud¯i, the following rule is given: The sum of the natural series being multiplied by eight and then increased by one, its square root decreased by one and halved, is the number of terms. Provide an algebraic justification for this rule.

Solution

Sum of natural number = 1 + 2 + 3 + 4 + 5 ......... + (N-1) + N = N*(N + 1)/2

Next

= 8*(1 + 2 + 3 + 4 + 5 ......... + (N-1) + N) = 8*(N^2 + N)/2 = 4N^2 + 4N

Next

= 8*(1 + 2 + 3 + 4 + 5 ......... + (N-1) + N) + 1 = 4N^2 + 4N + 1 = (2N+1)^2

Next

= sqrt[8*(1 + 2 + 3 + 4 + 5 ......... + (N-1) + N) + 1] = sqrt (2N + 1)^2 = 2N + 1

Next

= sqrt[8*(1 + 2 + 3 + 4 + 5 ......... + (N-1) + N) + 1] - 1 = 2N + 1 - 1 = 2N

final sequence will be

= [sqrt{8*(1 + 2 + 3 + 4 + 5 ......... + (N-1) + N) + 1} - 1]/2 = 2N/2 = N

Where N is number of terms.