The sum upto (2n+1) terms of the series a2−(a+d)2+(a+2d)2−(a+3d)2+… is

We can write the sum upto (2n+1) terms as[a+(a+d)](−d)+[(a+2d)+(a+3d)](−d)+…(a+(2n−2)d)+(a+(2n−1)d](−d)+(a+2nd)2          =(−d)[a+(a+d)+(a+2d)+…      +a+(2n−1)d]+(a+2nd)2= (−d)2n2{a+a+(2n−1)d}+(a+2nd)2=−2nad−n(2n−1)d2+a2+4n(ad)+4n2d2=a2+2nad+n(2n+1)d2=(a+nd)2+n(n+1)d2