⁻¹ (2 1² )+ ⁻¹ (2 2² )+ ⁻¹ (2 3² )+ upto is equal to

Step-by-step solution

⁻¹ (2 1² )+ ⁻¹ (2 2² )+ ⁻¹ (2 3² )+ = _ r =1 ^ ⁻¹ (2 r ² )= _ r =1 ^ ⁻¹ ( 1 2 r ² )= _ r =1 ^ ⁻¹ ( (1+2 r )+(1-2 r ) 1-(1+2 r )(1-2 r ) )= _ r =1 ^ [ ⁻¹(1+2 r )+ ⁻¹(1-2 r ) ]= ⁻¹ 3- ⁻¹ 1+ ⁻¹ 5- ⁻¹ 3+ ⁻¹ 7- ⁻¹ 5+ + ⁻¹ =- 4 + 2 = 4