Mean of the numbers 1, 2, 3,……,n with respective weights 12+1,22+2,32+3........,n2+n is

Here for each xi=i ,weight wi=i2+i Hence, the required mean =∑wixiwi=∑i=1ni(i2+i)∑i=1n(i2+i)                                                                  =∑i=1ni3+∑i=1ni2∑i=1ni2+∑i=1ni                                                                 =n2(n+1)24+n(n+1)(2n+1)6n(n+1)(2n+1)6+n(n+1)2                                                                 =n(n+1)2{n(n+1)2+2n+13}n(n+1)2{2n+13+1}                                                                =3n2+7n+22(2n+4)                                                                =(3n+1)(n+2)4(n+2)=3n+14