If S1 is the sum of an arithmetic series of ‘n’ odd number of terms and S2, the sum of the terms of the series in odd places, then S1S2=

Let the odd number of terms of an arithmetic series be a,a+d,a+2d,a+3d,a+4d,……,a+(n−1)dThen S1=n2{2a+(n−1)d}S2=a+(a+2d)+(a+4d)+… to n+12 terms  =n+12×22a+n+12−1×2d =n+14(2a+(n−1)d)∴S1S2=2nn+1