Let X = \ x ∈ N: 1 ≤ x ≤ 17\ and Y = \ ax + b: x ∈ Xanda, b ∈ R, a...

$\dfrac{a\left(17 \times 18\right)}{17 \times 2} + b = 17$

$9a + b = 17$

$\sigma\left({X}^{2}\right) = \dfrac{∑ {x}^{2}}{n} - {\left(\dfrac{∑ x}{n}\right)}^{2}$

$= \dfrac{{1}^{2} + {2}^{2} + \ldots + {17}^{2}}{17} - {\left(\dfrac{1 + 2 + . . . + 17}{17}\right)}^{2}$

$= \dfrac{17 \cdot 18 \cdot 35}{6 \cdot 17} - {\left(\dfrac{17 \cdot 18}{2 \cdot 17}\right)}^{2}$

$= 105 - 81 = 24$

$∴ \sigma\left({Y}^{2}\right) = {a}^{2}\sigma\left({X}^{2}\right) = {a}^{2} \cdot 24 = 216$

$⇒ {a}^{2} = \dfrac{216}{24} = 9$

$a = 3$

$∴ b = 17 - 27$

$∴ b = 17 - 27$

$b = - 10$

$∴ a + b = - 7$