The sum of the series 1+2+4+5+7+8+10+11+ ......+ 2999 is _____?

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Rewrite the series as

{1+4+7+10+...........+2998}+{2+5+8+11+.......+2999)

let sum of the series be

S={1+4+7+10+...........+2998}+{2+5+8+11+.......+2999)

Let

P=1+4+7+10+...........+2998

Q=2+5+8+11+.......+2999

Series P and Q are i A.P. with common difference 3 and first term 1 and 2 resp.

`a_n=1+(n-1)3=2998`

`(n-1)3=2997`

`n-1=999`

`=1000`

Thus P and Q has n=1000 terms.

`P=(1000/2)(2xx1+(1000-1)3)`

`=500(2+2997)=500(2999)=1499500`

`Q=(1000/2)(2xx2+(1000-1)3)=500(4+2999)=1501500`

Thus

**S=1499500+1501500=3001000**

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