If I_n= ₀^ / 4 ^n x d x , then I₂+I₄, I₃+I₅, I₄+I₆, , are in

Step-by-step solution

I_n= ₀^ / 4 ^n x d x We have, I_ r+2 = ₀^ / 4 ^ r+2 x d x= ₀^ / 4 ^r x ^2 x d x and I_r= ₀^ / 4 ^r x d x Then, I_r+I_ r+2 = ₀^ / 4 ^r x d x+ ₀^ / 4 ^r x ^2 x d x= ₀^ / 4 ^r x (1+ ^2 x ) d x= ₀^ / 4 ^r x ^2 x d x [Put t= x d t= ^2 x d x ] = ₀^1 t^r d t [ t^ r+1 r+1 ]₀^1= ( 1 r+1 ) So, I_r+I_ r+2 = 1 r+1 I₂+I₄= 1 3 I₃+I₅= 1 4 I₄+I₆= 1 5 which are clearly in HP.