Ankita has to climb 5 stairs starting at the ground, while respecting the following rules:
1. At any stage, Ankita can move either one or two stairs up.
2. At any stage, Ankita cannot move to a lower step.
Let $F(N)$ denote the number of possible ways in which Ankita can reach the $N^{th}$ stair. For example, $F(1) = 1$, $F(2) = 2$, $F(3) = 3$. The value of $F(5)$ is ________.
Problem Understanding: Stair Climbing Ways
Ankita needs to climb 5 stairs. She starts at the ground (step 0). At each step, she can either climb 1 stair or 2 stairs up. She cannot move down. We need to find the total number of distinct ways ($F(5)$) she can reach the 5th stair.
Defining the Recurrence Relation
Let $F(N)$ be the number of ways to reach the $N^{th}$ stair. To reach the $N^{th}$ stair, Ankita must have come from either the $(N-1)^{th}$ stair (by taking a single step) or the $(N-2)^{th}$ stair (by taking two steps). Therefore, the total number of ways to reach the $N^{th}$ stair is the sum of the ways to reach the $(N-1)^{th}$ stair and the ways to reach the $(N-2)^{th}$ stair.
The recurrence relation is: $F(N) = F(N-1) + F(N-2)$
Calculating F(5) Step-by-Step
We are given the values for the first few stairs:
- $F(1) = 1$ (Only one way: 1)
- $F(2) = 2$ (Two ways: 1+1, 2)
- $F(3) = 3$ (Three ways: 1+1+1, 1+2, 2+1)
Now, we use the recurrence relation to find $F(4)$ and $F(5)$:
- Calculate F(4): Using the relation $F(N) = F(N-1) + F(N-2)$: $F(4) = F(3) + F(2)$ $F(4) = 3 + 2$ $F(4) = 5$
- Calculate F(5): Using the relation again: $F(5) = F(4) + F(3)$ $F(5) = 5 + 3$ $F(5) = 8$
Thus, there are 8 possible ways for Ankita to reach the 5th stair.
Important Questions from Numerical Reasoning
- $P, Q, R, S, X$, and $Y$ are distinct single-digit whole numbers taking values from 0 to 9.
$PQ$ is a two-digit number with $Q$ being in the units place and $P$ in the tens place. Similarly, $RS$ is a two-digit number.
It is known that $PQ$ and $RS$ are consecutive numbers and
$(PQ)^2 + (RS)^2 = XYP$, with $XYP$ being a three-digit number.
The value of $Y$ is __________ - Let $p_1$ and $p_2$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_1$ and $p_2$?
If $\oplus \div \odot = 2$, $\oplus \div \triangle = 3$, $\odot + \triangle = 5$, and $\Delta \times \otimes = 10$,
then the value of $(\otimes - \oplus)^2$ is:- The remainder when $98!$ is divided by $101$ is equal to ________
- A 'frabjous' number is defined as a 3 digit number with all digits odd, and no two adjacent digits being the same. For example, 137 is a frabjous number, while 133 is not. How many such frabjous numbers exist?
