KVPY2012MathematicsSequences and Series
Consider array l L = [3] 2012 + [3] 2013 + + [3] 3011 R = [3] 2013 + [3] 2014 + + [3] 3012 array and I= ₂₀₁₂³⁰¹² [3] x d x Then -
Options
- AL + R < 2 I
- BL + R >2 I
- CL + R =2 I
- DLR = I
Correct answer
C. L + R =2 I
Step-by-step solution
L = [3] 2012 + [3] 2013 + + [3] 3011 (1) R = [3] 2013 + [3] 2014 + + [3] 3012 (2) I = ₂₀₁₂³⁰¹² x ^ 1 / 3 dx Let f(x)=x^ 1 / 3 n = b - a h = 3012-2012 1 =1000 I = ( b - a ) n [ f ( a )+ f ( a + h )+ f ( a +2 ~h )+ + f ( a +( n -1) h )]=[ f (2010)+ f (2013)+ + f (3011)] I =(2012)^ 1 / 3 +(2013)^ 1 / 3 + . .+(3011)^ 1 / 3 2 I =2(2012)^ 1 / 3 +2(2013)^ 1 / 3 + . .+2(3011)^ 1 / 3 =(2012)^ 1 / 3 +(2012)^ 1 / 3 +2(2013)^ 1 / 3 + .+(2)(3011)^ 1 / 3 +(3012)^ 1 / 3 -(3012)^ 1 / 3 =(2012)^ 1 / 3 + L + R -(3012)^ 1 / 3 2 I < L