Prove That X ⊆ Y
If \[ X=\{8^n-7n-1:n \in N\} \] and \[ Y=\{49(n-1):n \in N\} \] then prove that \[ X \subseteq Y \]
Solution
Let \[ x=8^n-7n-1 \] where \[ n \in N \]
Using binomial expansion, \[ 8^n=(7+1)^n \]
\[ =(1+n \cdot 7+\text{terms containing }49) \]
Therefore, \[ 8^n-7n-1 \] is divisible by \[ 49 \]
Hence, \[ 8^n-7n-1=49k \] for some \[ k \in N \]
So, \[ x \in Y \]
Therefore, \[ \boxed{X \subseteq Y} \]