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MHT CET20244 May 2024Evening ShiftMathematicsContinuity and DifferentiabilityActual

If f (x)= x+x^2+x^3+ +x^ n - n x-1 , for x 1 is continuous at x=1 , then f (1)=

Options

  1. An ( n +1)(4 n -1) 6
  2. Bn ( n +1) 2
  3. Cn ( n +1)(2 n +1) 6
  4. Dn (2 n +1) 4

Correct answer

B. n ( n +1) 2

Step-by-step solution

aligned f (1) & = _ x 1 f (x) & = _ x 1 x+x^2+x^3+ +x^ n - n x-1 & = _ x 1 x-1+x^2-1+x^3-1+ +x^ n -1 x-1 aligned aligned & = _ x 1 x-1 x-1 + _ x 1 x^2-1 x-1 + + _ x 1 x^ n -1 x-1 & =1+2(1)²⁻¹+3(1)³⁻¹+ + n (1)^ n -1 & =1+2+3+ n f (1) & = n ( n +1) 2 aligned

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