Dn Recursion Ded11A
Let's explore the formula step by step! 🎉
1. Imagine you have some numbers called $D_n$ and $D_{n-1}$.
2. The formula says: $$D_n = nD_{n-1} + (-1)^n$$
3. The $(-1)^n$ part means:
- If $n$ is even, $(-1)^n = 1$.
- If $n$ is odd, $(-1)^n = -1$.
4. Let's look at some examples using candies 🍬:
**Box 1:**
Group 1 (for $D_{n-1}$):
🍬 🍬
(2 candies)
Add ➕
**Box 2:**
🍬 🍬
(2 candies)
Group 2 (for $(-1)^n$ when $n=3$):
❌
(-1 candy means take one away)
=
**Total box:**
❌
(-1 candy means take one away)
Total $D_n$:
🍬 🍬 🍬 🍬 🍬
(5 candies)
5. So for $n=3$, $D_3 = 3 imes D_2 + (-1)^3 = 3 imes 2 - 1 = 5$.
Great! This formula helps you find $D_n$ using the last number $D_{n-1}$ and a little +1 or -1 twist depending on if $n$ is even or odd. 🌟
Keep practicing and you will be a math star! 🚀🍬 🍬 🍬 🍬 🍬
(5 candies)