find the valueof (99^3+100^3)/(1^3+100^3)+(97^3+99^3)/(2^3+99^3)+(95^3+98^3)/(3^3+98^3)+...+1^3+51^3)/(50^3+51^3)?

Sol
m=101
A=(99^3+100^3)/(1^3+100^3)+(97^3+99^3)/(2^3+99^3)+(95^3+98^3)/(3^3+98^3)
+...+(1^3+51^3)/(50^3+51^3)
=(99^3+100^3)/(1^3+100^3)+(97^3+99^3)/(2^3+99^3)+(95^3+98^3)/(3^3+98^3)+...
+(1^3+51^3)/(50^3+51^3)
=Σ(k=1 to 50)_｛[(101－2k)^3+(101－k)^3]/[k^3+(101－k)^3]｝
=Σ(k=1 to 50)_｛[(m－2k)^3+(m－k)^3]/[k^3+(m－k)^3]｝
(m－2k)^3+(m－k)^3
=(m^3－3*m^2*2k+3*m*4k^2－8k^3)+(m^3－3m^2k+3mk^2－k^3)
=2m^3－9m^2k+15mk^2－9k^3
=(2m－3k)(m^2－3mk+3k^2)
k^3+(m－k)^3=m^3－3m^2k+3mk^2=m(m^2－3mk+k^2)
A=Σ(k=1 to 50)_｛[(m－2k)^3+(m－k)^3]/[k^3+(m－k)^3]｝
=Σ(k=1 to 50)_(2m－3k)/m
=100－(3/m)*Σ(k=1 to 50)_k
=100-(3/101)*50*51/2
=6275/101