Ant Movement Calculation
The problem asks for the coordinates of an ant after completing 5 stages of movement, starting from the origin.
Stage-by-Stage Coordinate Tracking
The ant's movement follows a pattern: initial movement along the y-axis, followed by subsequent stages involving a right turn and movement covering half the previous distance.
- Stage 1: Starts at origin $(0, 0)$. Moves along the positive $y$-axis for distance $l$.
- Coordinates: $(0, l)$
- Direction: Positive $y$-axis
- Stage 2: Turns right from positive $y$-axis (moves along positive $x$-axis). Distance is $l/2$.
- Starting coordinates: $(0, l)$
- Movement: $(l/2, 0)$
- Final coordinates: $(0 + l/2, l) = (\frac{l}{2}, l)$
- Direction: Positive $x$-axis
- Stage 3: Turns right from positive $x$-axis (moves along negative $y$-axis). Distance is $(l/2)/2 = l/4$.
- Starting coordinates: $(\frac{l}{2}, l)$
- Movement: $(0, -l/4)$
- Final coordinates: $(\frac{l}{2}, l - \frac{l}{4}) = (\frac{l}{2}, \frac{3l}{4})$
- Direction: Negative $y$-axis
- Stage 4: Turns right from negative $y$-axis (moves along negative $x$-axis). Distance is $(l/4)/2 = l/8$.
- Starting coordinates: $(\frac{l}{2}, \frac{3l}{4})$
- Movement: $(-l/8, 0)$
- Final coordinates: $(\frac{l}{2} - \frac{l}{8}, \frac{3l}{4}) = (\frac{4l}{8} - \frac{l}{8}, \frac{3l}{4}) = (\frac{3l}{8}, \frac{3l}{4})$
- Direction: Negative $x$-axis
- Stage 5: Turns right from negative $x$-axis (moves along positive $y$-axis). Distance is $(l/8)/2 = l/16$.
- Starting coordinates: $(\frac{3l}{8}, \frac{3l}{4})$
- Movement: $(0, l/16)$
- Final coordinates: $(\frac{3l}{8}, \frac{3l}{4} + \frac{l}{16}) = (\frac{3l}{8}, \frac{12l}{16} + \frac{l}{16}) = (\frac{3l}{8}, \frac{13l}{16})$
- Direction: Positive $y$-axis
Final Coordinates
At the end of the 5th stage, the ant's coordinates are $(\frac{3l}{8}, \frac{13l}{16})$.
Important Questions from Progression (Notes)
- The sum of 16 terms of the series $\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} + .....$ is :
- A pilgrim starts walking for a journey of 115 km. On the first day he covers 7 km, the next day 9 km, and likewise keeps adding 2 km everyday till he reaches 15 km per day which he maintains for the rest of the journey. How many days in all will he take to complete the journey?
If a, b and c are in Geometric Progression and $a^\frac{1}{x} = b^\frac{1}{y} = c^\frac{1}{z}$ then, x, y, z are in ________.
Which of the following statement is true about the geometric series
$ 1 + r +r^2 + r^3 + ...............; (r > 0) $?
$6240$ रुपये की राशि $30$ किस्तों में इस प्रकार चुकाई जाती है कि प्रत्येक किस्त पिछली किस्त से $10$ रुपये अधिक है । पहली किस्त की मूल्य ____________है।
