Describe the pattern of the sequence, $-1,\ 8,\ 23,\ 44$−1, 8, 23, 44​ .The pattern is .Question 2Find the next term.The next term is .

1. Break Down the Problem

To identify the pattern of the given sequence $-1, 8, 23, 44$, we first need to analyze the differences between successive terms.

2. Relevant Concepts

The differences between terms in a sequence can reveal a pattern. We will find the first-order differences by subtracting each term from the following term. Then, we can look for any pattern in the differences or their further differences.

3. Analysis and Detail

1. Calculate the first-order differences:

   • $8 - (-1) = 9$
   • $23 - 8 = 15$
   • $44 - 23 = 21$

   The first-order difference sequence is: $9, 15, 21$.

2. Calculate the second-order differences:

   • $15 - 9 = 6$
   • $21 - 15 = 6$

   The second-order difference sequence is constant: $6$.

This indicates that the original sequence is quadratic since the second-order differences are constant.

4. Verify and Summarize

The quadratic form can be expressed as: $a_n = An^2 + Bn + C$ To find $A$, $B$, and $C$, we can set up the following system of equations based on the known terms:

1. For $n=1$ (term is $-1$): $A(1)^2 + B(1) + C = -1$ $A + B + C = -1 \quad \text{(Equation 1)}$

2. For $n=2$ (term is $8$): $A(2)^2 + B(2) + C = 8$ $4A + 2B + C = 8 \quad \text{(Equation 2)}$

3. For $n=3$ (term is $23$): $A(3)^2 + B(3) + C = 23$ $9A + 3B + C = 23 \quad \text{(Equation 3)}$

Now we can solve this system of equations:

From Equation 1: $C = -1 - A - B$

Substituting $C$ into Equation 2: $4A + 2B + (-1 - A - B) = 8$ $3A + B = 9 \quad \text{(Equation 4)}$

Substituting $C$ into Equation 3: $9A + 3B + (-1 - A - B) = 23$ $8A + 2B = 24 \quad \text{(Equation 5)}$

Now, simplify Equation 5: $4A + B = 12 \quad \text{(Equation 6)}$

Now, we have two equations:

1. $3A + B = 9$ (Equation 4)
2. $4A + B = 12$ (Equation 6)

Subtract Equation 4 from Equation 6: $(4A + B) - (3A + B) = 12 - 9$ $A = 3$

Substituting $A = 3$ back into Equation 4: $3(3) + B = 9$ $B = 0$

Now substituting $A$ and $B$ back into Equation 1 to find $C$: $3 + 0 + C = -1$ $C = -4$

Thus, the formula for the sequence is: $a_n = 3n^2 - 4$

Now, we can find the next term ($n=5$): $a_5 = 3(5)^2 - 4 = 3 \cdot 25 - 4 = 75 - 4 = 71$

Final Answer

The next term is $71$.