x 3 + 1, x 4 + 2, x 5 + 3 ......

+ 4 x 2, + 8 x 3, + 12 x 4

The series is based on combination of two series. S₁ = + 13, +26, +39 ...... and S₂ = +7, +14, +21 .......

+36, +72, +144, +288 .......

(22)², (27)², (32)², (37)² .........

I. x² + 7x - 4x - 28 = 0

or, x(x + 7) - 4 (x + 7) = 0

or, (x - 4) (x + 7) = 0

∴ x = 4, -7

II. y² - 11y + 28 = 0

or, y2 - 7y - 4y + 28 = 0

or, y (y - 7) - 4 (y - 7) = 0

or, (y - 4) (y - 7) = 0

∴ y = 4, 7

∴ x < y

I. 6x² - 17x + 12 = 0

or, 6x² - 9x - 8x + 12 = 0

or, 3x (2x - 3) - 4 (2x - 3) = 0

or, (3x - 4) (2x - 3) = 0

∴ x = 4/3, 3/2

II. 6y² - 3y - 4y + 2 = 0

or, 3y (2y - 1) - 2 (2y - 1) = 0

or, (3y - 2) (2y - 1) = 0

∴ y = 2/3, 1/2

∴ x > y

I. x = √256/√576

∴ x = 16/24 = 2/3

II. 3y² + y - 2 = 0

or, 3y² + 3y - 2y - 2 = 0

or, 3y (y + 1) - 2 (y + 1) = 0

or, (3y - 2) (y + 1) = 0

∴ y = 2/3, -1

∴ x > y

I. x² = 64

∴ x = + 8

II. y² = 9y

or, y² - 9y = 0

or, y (y - 9) = 0

∴ y = 0, 9

∴ no relationship can be established between x and y.

I. x² + 6x - 7 = 0

or, x² + 7x - 7 = 0

or, x(x + 7) - 1 (x + 7) = 0

or, (x - 1) (x + 7) = 0

∴ x = 1, -7

II. 41y + 17 = 140

or, 41y = 140 - 17 = 123

∴ y = 123/41 = 3

∴ x < y