Question

If X = {8ⁿ – 7n – 1 : n ∈ N} and Y = {49n – 49 : n ∈ N}. Then,
(a) X ⊂ Y
(b) Y ⊂ X
(c) X = Y
(d) X ∩ Y = ϕ

Answer 1

$$\begin{gathered} X=\left\{8^n-7n-1,n\in N\right\} \\ Y=\left\{49n-49;n\in N\right\} \\ \mathrm{Since}{8}^n-7n-1={\left(1+7\right)}^n-7n-1 \\ =\left(1+7n{+}^n{C}_2{7}^2{+}^n{C}_3{7}^3+....+7^n\right)-7n-1 \\ =1+7n{+}^n{C}_{2}49{+}^n{C}_3{7}^3+....+7^n-7n-1 \\ =7^2\left({}^n{C}_2{+}^n{C}_{3}7+....+7^{n-2}\right) \\ =49\left({}^n{C}_2{+}^n{C}_{3}7+...+7^{n-2}\right) \\ \mathrm{i}.\mathrm{e}.8^n-7n-1\mathrm{is}\mathrm{a}\mathrm{multiple}\mathrm{of}49 \\ \mathrm{i}.\mathrm{e}.X\subset Y \\ \mathrm{Here}\mathrm{X}=\left\{0,49,490,4067,....\right\} \\ Y=\left\{0,49,98,.....\right\} \\ Y\not\subset X\left(\mathrm{since}98\notin x\right) \\ \therefore X\subset Y \end{gathered}$$

Hence, the correct answer is option A.