sn=1+3+5+7+.... which is equal to sum of odd term series.... s1=(1*1)-(0-0)=1; s2=(2*2)-(1*1)=3; s3=(3*3)-(2*2)=5; s4=(4*4)-(3*3)=7; from above s1,s2,s3,s4 equations we can see that first term of first equation(s1) and second term of second equation(s2) gets cancelled,simillarly first term of second equation(s2) and second term of thid equation(s3) gets cancelled..and so on... at last we remain with (4*4)-(0*0) i.e (n*n)-(0*0) i.e (n*n) therefore... sn=n*n; given in question that sn%(10^9+7) ->(n*n)%(10^9+7) we have a theorem which states that.. (a*b)%n=((a%n)*(b%n))%n ->(n*n)%(10^9+7) ->(n*n)%(1000000007) frm above theorem.. ->((n%1000000007)*(n%1000000007))%1000000007) the above theorem helps us to clear test case 5 & 6