If the sum of all the positive even integers less than 1,000,000 is equal to W, then what is the sum of all of the positive odd integers less than 1,000,000?
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Sum of n terms of an Arithmetic Progression where a and l are first and last terms respectively is -
2 n × ( a + l )
Given that the sum of the Arithmetic Progression 2,4,6,...,999998 is W,
W = 2 n × ( 2 + 9 9 9 9 9 8 )
⟹ n = 1 , 0 0 0 , 0 0 0 2 W
Now, there will be 'n+1' terms in the AP of odd numbers less than 1,000,000. Let the sum of odd numbers less than 1,000,000 be S.
S = 2 n + 1 × ( 1 + 9 9 9 9 9 9 )
Substituting value of n in the above equation,
S = ( 1 0 0 0 0 0 0 W + 2 1 ) × ( 1 0 0 0 0 0 0 )
∴ S = W + 5 0 0 , 0 0 0