Question

If $S_r=\sqrt{r+\sqrt{r+\sqrt{r+...........upto\infty }}}(r>0)$,then which is correct.

• A. $S_2,S_6,S_{12},S_{20}......$ are in A.P.
• B. $S_4,S_9,S_{16}$ are irrational.
• C. $(2S_2-1{)}^2,(2S_3-1{)}^2,(2S_4-1{)}^2$ are in A.P.
• D. all of these

Answer 1

Answer: (D)

all of these

$S=\sqrt{r+\sqrt{r+\sqrt{r+\sqrt{r+...}}}}=>S^2=r+S=>S^2-S-r=0=>S=\frac{1\pm \sqrt{1+4r}}{2}=\frac{1+\sqrt{1+4r}}{2}\left(i\right)S_2=2,S_6=3,S_{12}=4,S_{20}=5=>TheyareinAP\left(ii\right)S_4=\frac{1+\sqrt{17}}{2},S_9=\frac{1+\sqrt{37}}{2},S_{16}=\frac{1+\sqrt{65}}{2}=>Theyareirrational\left(iii\right){(2S_2-1)}^2=9,{(2S_3-1)}^2=13,{(2S_4-1)}^2=17=>TheyareinAP$
Hence, all are true.
Hence, (d) is correct.