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BITSAT2024MathematicsSequences and SeriesActual

If y= ⁻¹ 1 x^2+x+1 + ⁻¹ 1 x^2+3 x+3 + ⁻¹ 1 x^2+5 x+7 + .to n terms, then d y d x =

Options

  1. A1 x^2+n^2 - 1 x^2+1
  2. B1 (x+n)^2+1 - 1 x^2+1
  3. C1 x^2+(n+1)^2 - 1 x^2+1
  4. DNone of these

Correct answer

B. 1 (x+n)^2+1 - 1 x^2+1

Step-by-step solution

Given, y= ⁻¹ 1 x^2+x+1 + ⁻¹ 1 x^2+3 x+3 + ⁻¹ 1 x^2+5 x+7 + to n terms = ⁻¹ 1 1+x(x+1) + ⁻¹ 1 1+(x+1)(x+2) + ⁻¹ 1 1+(x+2)(x+3) + + ⁻¹ 1 1+(x+(n-1))(x+n) = ⁻¹ (x+1)-x 1+(x+1) x + ⁻¹ (x+2)-(x+1) 1+(x+2)(x+1) + ⁻¹ (x+3)-(x+2) 1+(x+3)(x+2) + .+ ⁻¹ ( (x+n)-(x+n-1) 1+(x+n)(x+n-1) ) y= ⁻¹(x+1)- ⁻¹(x) + ⁻¹(x+2)- ⁻¹(x+1) + ⁻¹(x+3)- ⁻¹(x+2) + . .+ ⁻¹(x+n)- ⁻¹[x+(n-1)] So, y= ⁻¹(x+n)- ⁻¹(x) On differentiating both sides w.r.t. x , we get d y d x = 1 1+(x+n)^2 - 1 1+x^2

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