Question
For any set A, (A')' is equal to:
- A'
- A
- $\phi$
- None of these.
Solution:
The complement of the complement of a set is the set itself.
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$a_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}, n \geq 1$
$b_1=1 \text { and } b_n=a_{n-1}+a_{n+1}, n \geq 2.$
Then which of the following options is/are correct?
$(1)$ $a_1+a_2+a_3+\ldots . .+a_n=a_{n+2}-1$ for all $n \geq 1$
$(2)$ $\sum_{n=1}^{\infty} \frac{ a _{ n }}{10^{ n }}=\frac{10}{89}$
$(3)$ $\sum_{n=1}^{\infty} \frac{b_n}{10^n}=\frac{8}{89}$
$(4)$ $b=\alpha^n+\beta^n$ for all $n>1$