If
\[ X=\{8^n-7n-1:n\in N\} \]
and
\[ Y=\{49n-49:n\in N\} \]
Then,
(a) \(X\subset Y\)
(b) \(Y\subset X\)
(c) \(X=Y\)
(d) \(X\cap Y=\phi\)
Solution
Consider
\[ 8^n=(1+7)^n \]
Using binomial expansion,
\[ 8^n=1+7n+\text{terms containing }7^2 \]
Therefore,
\[ 8^n-7n-1 \]
is divisible by
\[ 49 \]
Hence every element of \(X\) is a multiple of \(49\).
Now,
\[ Y=\{49(n-1):n\in N\} \]
which is also the set of multiples of \(49\).
Therefore,
\[ X=Y \]
Answer
\[ \boxed{X=Y} \]
Correct option: (c)