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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 $n>1$
$f_1(x)=\left\{\begin{array}{lll}|x| & \text { if } & x<0, \\ e^x & \text { if } & x \geq 0 ;\end{array}\right.$
$f_2(x)=x^2$
$f_3(x)=\left\{\begin{array}{ccc}\sin x & \text { if } & x < 0, \\ x & \text { if } & x \geq 0\end{array}\right.$ and
$f_4(x)=\left\{\begin{array}{ccc}f_2\left(f_1(x)\right) & \text { if } & x < 0, \\ f_2\left(f_1(x)\right)-1 & \text { if } & x \geq 0\end{array}\right.$
| List $I$ | List $II$ |
| $P.$ $ f_4$ is | $1.$ onto but not one-one |
| $Q.$ $f_3$ is | $2.$ neither continuous nor one-one |
| $R.$ $f _2 \circ f _1$ is | $3.$ differentiable but not one-one |
| $S.$ $ f_2$ is | $4.$ continuous and one-one |
Codes: $ \quad P \quad Q \quad R \quad S $