A line segment initially 1 unit long grows according to the law 1+(√2/4)+(1/4)+(√2/16)+(1/16)+(√2/64)+(1/64)+ ldots ldots . . . ∞ If the growth process continues indefinitely, then length of the line is equal to

Question

A line segment initially 1 unit long grows according to the law 1+(√2/4)+(1/4)+(√2/16)+(1/16)+(√2/64)+(1/64)+ ldots ldots . . . ∞ If the growth process continues indefinitely, then length of the line is equal to (A) (4/3) (B) (8/3) (C) (4+√2/3) (D) (2/3)(4+√2)

Tardigrade Admin · Accepted Answer

S =(1+(1/4)+(1/16)+(1/64)+ ldots ldots ∞)+((√2/4)+(√2/16)+(√2/64)+ ldots ldots ∞) S =(1/1-(1)4)+((√2/4)/1-(1)4)=(4/3)+(4/3)((√2/4))=(4+√2/3)

Q. A line segment initially 1 unit long grows according to the law
$1 + \frac{2}{4} + \frac{1}{4} + \frac{2}{16} + \frac{1}{16} + \frac{2}{64} + \frac{1}{64} + \dots\dots ...\infty$
If the growth process continues indefinitely, then length of the line is equal to

$\frac{4}{3}$

$\frac{8}{3}$

$\frac{4 + 2}{3}$

$\frac{2}{3} (4 + 2)$

$S = (1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \dots\dots \infty) + (\frac{2}{4} + \frac{2}{16} + \frac{2}{64} + \dots\dots \infty)$
$S = \frac{1}{1 - \frac{1}{4}} + \frac{\frac{2}{4}}{1 - \frac{1}{4}} = \frac{4}{3} + \frac{4}{3} (\frac{2}{4}) = \frac{4 + 2}{3}$

Q. A line segment initially 1 unit long grows according to the law 1+42​​+41​+162​​+161​+642​​+641​+……...∞ If the growth process continues indefinitely, then length of the line is equal to

34​

38​

34+2​​

32​(4+2​)