Find the sum of the following series :
`5+(-41)+9+(-39)+13+(-37)+17+....+(-5)+81+(-3)`

Explanation

Let `S=5+(-41)+9+(-39)+13+(-37)+17+...+(-5)+81+(-3)`
`=(5+9+13+17+...81)-(41+39+37+...+3)`
`S_(1)-S_(2) " "...(1)`
where, `S_(1)=5+9+13+17+...+81`
It is an A.P. with `a_(1)=5, d=9-5=4`
Let these are n terms.
`:. a_(n)=81`
`rArr a_(1)+(n-1)d=81`
`rArr5+(n-1)4=81`
`rArr(n-1)4=76 rArr n-1=19 rArr n=20`
`:. S_(1)`=Sum of 20 terms with term 5 and last term 81
`=(20)/(2)[5+81]=10xx86=860 " " ...(2)`
and `S_(2)=41+39+37+...+3`
It is also an A.P. with `a_(1)=41,d=39-41=-2`
Let these are n terms.
`:. a_(n)=3`
`rArr a_(1)+(n-1)d=3`
`rArr 41+(n-1)(-2)=3`
`rArr (n-1)(-2)=-38`
`rArr n-1=19 rArr n=20`
`:. S_(2)`=Sum of 20 terms with first term 41 and last term 3
`=(20)/(2)[41+3]=10xx44=440 " " ...(3)`
`:.` From equation (1),
`S=S_(1)-S_(2)`
`rArr S=860-440=420`