Mathematics in Ancient and Medieval India c7.s2.42

c7.s2.42

Reference may be made here that the Jain mathematician Mahāvīra (850 A.D.) also gave the results of the generalised (algebraic) summation of the square and cube of the series a, a+b, a+2b,......corresponding to Āryabhaṭa I's work in natural number as follows : $a² + (a+b)² + (a+2b)² + .........to n terms$ $= n [((2n-1) b²)/b + ab (n-1) + a²]$ and a³ + (a+b)³ + (a+2b)³ + ......to n terms $= S²b + Sa (a-b) if a > b$ $= S²b - Sa (a-b) if a < b$ where S = a + (a+b) + (a+2b) + ......to n terms.