Question

Make a conjecture for the next equation in the pattern and verify it. The pattern is based on the Fibonacci Sequence. The given equations are:

1!+1!=2

1!+2!=5

2!+3!=13

3!+5!=34
Conjecture for the next equation:
n!+(n−1)!=?

Answer 1

The pattern is based on the Fibonacci Sequence. To find the next equation, evaluate n! + (n-1)!. This conjecture can be verified using the given examples.

Step-by-step explanation:

The pattern is based on the Fibonacci Sequence. In the given equations, we can observe that the first term is the factorial of the current number, n, and the second term is the factorial of the previous number, n-1. To find the next equation, we need to evaluate n! + (n-1)!. For example, if the current term is 4, the equation would be 4! + 3!.

Let's verify this conjecture using the given examples:

1. 1! + 1! = 2
2. 1! + 2! = 5
3. 2! + 3! = 13
4. 3! + 5! = 34

Plugging in the numbers, we get:

• 1! + 1! = 1 + 1 = 2
• 1! + 2! = 1 + 2 = 3
• 2! + 3! = 2 + 6 = 8
• 3! + 5! = 6 + 120 = 126

Therefore, the next equation in the pattern would be n! + (n-1)! = n! + ((n-1)!).