Let $P =\sum_{\substack{1\le i \le 2k \\ i\;odd}} i$ and $Q = \sum_{\substack{1 \le i \le 2k \\ i\;even}} i$, where $k$ is a positive integer. Then
P=1+3+5+7+........+(2k-1) =(2-1)+(4-1)+(6-1)+(8-1)+..........+(2k-1) =(2+4+6+8+.....2k)+(-1-1-1-1-1.....k times) =Q+(-k)=Q-k
Nice approach.Thanks :-)
Just small correction needed."The odd series is 2 4 6 8 10 ... 2k ."
Gatecse