If the sequence a_{1},a_{2},a_{3},a_{4},......a_{n}form an A.P. Then
value of a_{1}^{2} - a_{2}^{2} + a_{3}^{2}
-a_{4}^{2} +.....+a_{2n-1}^{2}
- a_{2n}^{2} is
[n/(2n-1)] [ a_{1}^{2}
- a_{2n}^{2}] . If we rearrange the terms-(a_{1}^{2} - a_{2}^{2}
)+ (a_{3}^{2}
-a_{4}^{2} )+.....+(a_{2n-1}^{2}
- a_{2n}^{2} ) which are up to n terms. Each
expression under bracket can be put as A1+A2+A3+...._An where A1=(a_{1}^{2} - a_{2}^{2}
), A2=(a_{3}^{2}
-a_{4}^{2} ) etc and A1,A2,..... are in AP &
common difference=square root of (A1-A2)/4. |