If x > 0 and x1 + x-1 ≤ 2, then what is the value of  (x1 + x -1)1 + (x2 + x-2)2 + (x3 + x-3)3 + ⋯ (x11 + x -11)11 is :

If x > 0 and x1 + x-1 ≤ 2, then what is the value of  (x1 + x -1)1 + (x2 + x-2)2 + (x3 + x-3)3 + ⋯ (x11 + x -11)11 is : Correct Answer 4094

Given:

x¹ + x^-1 ≤ 2       ----(1)

Calculation:

x + ( 1 / x ) ≤ 2

x² + 1 ≤ 2x

x2 - 2x + 1 ≤ 0

x2 - x - x + 1 ≤ 0

x ( x - 1 ) - 1 ( x - 1 ) ≤ 0

( x -1 ) ( x - 1 ) ≤ 0

So, x = 1

Subsituting in (i),

( 1 + 1 ) = 2

Thus,

(x + ( 1/x ) ) + ( x² + ( 1/x )² + .......+ ( x¹¹ + ( 1/x )¹¹ )

= ( 2 ) + ( 2 ) ² + ( 2 ) ³ + ........ + ( 2 ) ¹¹

The above equation is in geometric progression,

So, Sum =

Sum=

= 2 × ( 2048 - 1 )

= 2 × 2047

= 4094

∴ (x + ( 1/x ) ) + ( x2 + ( 1/x )2 + .......+ ( x11 + ( 1/x )11 ) = 4094