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Algebra Level 3

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?

W+500,000 W/2+999,999 W W+1

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1 solution

Sum of n terms of an Arithmetic Progression where a and l are first and last terms respectively is -

n 2 × ( a + l ) \frac{n}{2}\times(a + l)

Given that the sum of the Arithmetic Progression 2,4,6,...,999998 is W,

W = n 2 × ( 2 + 999998 ) W=\frac{n}{2}\times(2 + 999998)

n = 2 W 1 , 000 , 000 \implies n=\frac{2W}{1,000,000}

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 = n + 1 2 × ( 1 + 999999 ) S=\frac{n+1}{2}\times(1 + 999999)

Substituting value of n in the above equation,

S = ( W 1000000 + 1 2 ) × ( 1000000 ) S=(\frac{W}{1000000}+\frac{1}{2})\times(1000000)

S = W + 500 , 000 \therefore S = W+500,000

I have solved using sigma. :-)

Sachin Vishwakarma - 5 years, 8 months ago

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