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 :
- An auditorium has 8 seats in the first row, with every row to follow having 4 more seats than its preceding row. The total capacity is 416. What is the minimum number of rows needed to seat 150 people?
- The $5^{\text{th}}$ and $9^{\text{th}}$ terms of an arithmetic progression are 7 and 13 respectively. What is the $15^{\text{th}}$ term?
- Find the sum of the G.P.:
$5/11, 5/121, 5/1331, 5/14641, ...$ to $n$ terms. Suppose $a_1, a_2,..., a_{300}$ are integers such that $a_{i-1}+ a_i+ a_{i+1} = 2025$ for all $i = 2,3, ..., 299$.
If $a_7 = -5, a_9 = 37$, then the value of $a_{106}$ is
