Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Our Q&A platform offers a seamless experience for finding reliable answers from experts in various disciplines. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To find the nth term of the given quadratic sequence 3, 8, 15, 24, 35, ..., follow these steps:
### Step 1: Determine the Sequence Type
The first differences between consecutive terms help identify the sequence type:
[tex]\[ \begin{aligned} 8 - 3 &= 5 \\ 15 - 8 &= 7 \\ 24 - 15 &= 9 \\ 35 - 24 &= 11 \\ \end{aligned} \][/tex]
The first differences are: 5, 7, 9, 11.
Calculate the second differences:
[tex]\[ \begin{aligned} 7 - 5 &= 2 \\ 9 - 7 &= 2 \\ 11 - 9 &= 2 \\ \end{aligned} \][/tex]
The second differences are constant: 2. Thus, we have a quadratic sequence.
### Step 2: General Form of a Quadratic Sequence
The nth term of a quadratic sequence can be represented as:
[tex]\[ a_n = An^2 + Bn + C \][/tex]
### Step 3: Setup and Solve the Equations
We need to find the coefficients [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] given the first few terms of the sequence. Set up equations using the given terms:
For [tex]\(n = 1\)[/tex]:
[tex]\[ A(1)^2 + B(1) + C = 3 \][/tex]
[tex]\[ A + B + C = 3 \][/tex]
For [tex]\(n = 2\)[/tex]:
[tex]\[ A(2)^2 + B(2) + C = 8 \][/tex]
[tex]\[ 4A + 2B + C = 8 \][/tex]
For [tex]\(n = 3\)[/tex]:
[tex]\[ A(3)^2 + B(3) + C = 15 \][/tex]
[tex]\[ 9A + 3B + C = 15 \][/tex]
Now, we have the system of equations:
[tex]\[ \begin{aligned} 1. & \quad A + B + C = 3 \\ 2. & \quad 4A + 2B + C = 8 \\ 3. & \quad 9A + 3B + C = 15 \\ \end{aligned} \][/tex]
### Step 4: Solve the System of Equations
Subtract equation 1 from equation 2:
[tex]\[ (4A + 2B + C) - (A + B + C) = 8 - 3 \][/tex]
[tex]\[ 3A + B = 5 \quad \text{(Equation 4)} \][/tex]
Subtract equation 2 from equation 3:
[tex]\[ (9A + 3B + C) - (4A + 2B + C) = 15 - 8 \][/tex]
[tex]\[ 5A + B = 7 \quad \text{(Equation 5)} \][/tex]
Subtract equation 4 from equation 5:
[tex]\[ (5A + B) - (3A + B) = 7 - 5 \][/tex]
[tex]\[ 2A = 2 \][/tex]
[tex]\[ A = 1 \][/tex]
Substitute [tex]\(A = 1\)[/tex] back into equation 4:
[tex]\[ 3(1) + B = 5 \][/tex]
[tex]\[ 3 + B = 5 \][/tex]
[tex]\[ B = 2 \][/tex]
Finally, substitute [tex]\(A = 1\)[/tex] and [tex]\(B = 2\)[/tex] back into equation 1:
[tex]\[ 1 + 2 + C = 3 \][/tex]
[tex]\[ 3 + C = 3 \][/tex]
[tex]\[ C = 0 \][/tex]
### Step 5: Formulate the nth Term
With [tex]\(A = 1\)[/tex], [tex]\(B = 2\)[/tex], and [tex]\(C = 0\)[/tex], the nth term of the sequence is:
[tex]\[ a_n = n^2 + 2n \][/tex]
Thus, the nth term of the quadratic sequence [tex]\(3, 8, 15, 24, 35, \ldots\)[/tex] is:
[tex]\[ \boxed{a_n = n^2 + 2n} \][/tex]
### Step 1: Determine the Sequence Type
The first differences between consecutive terms help identify the sequence type:
[tex]\[ \begin{aligned} 8 - 3 &= 5 \\ 15 - 8 &= 7 \\ 24 - 15 &= 9 \\ 35 - 24 &= 11 \\ \end{aligned} \][/tex]
The first differences are: 5, 7, 9, 11.
Calculate the second differences:
[tex]\[ \begin{aligned} 7 - 5 &= 2 \\ 9 - 7 &= 2 \\ 11 - 9 &= 2 \\ \end{aligned} \][/tex]
The second differences are constant: 2. Thus, we have a quadratic sequence.
### Step 2: General Form of a Quadratic Sequence
The nth term of a quadratic sequence can be represented as:
[tex]\[ a_n = An^2 + Bn + C \][/tex]
### Step 3: Setup and Solve the Equations
We need to find the coefficients [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] given the first few terms of the sequence. Set up equations using the given terms:
For [tex]\(n = 1\)[/tex]:
[tex]\[ A(1)^2 + B(1) + C = 3 \][/tex]
[tex]\[ A + B + C = 3 \][/tex]
For [tex]\(n = 2\)[/tex]:
[tex]\[ A(2)^2 + B(2) + C = 8 \][/tex]
[tex]\[ 4A + 2B + C = 8 \][/tex]
For [tex]\(n = 3\)[/tex]:
[tex]\[ A(3)^2 + B(3) + C = 15 \][/tex]
[tex]\[ 9A + 3B + C = 15 \][/tex]
Now, we have the system of equations:
[tex]\[ \begin{aligned} 1. & \quad A + B + C = 3 \\ 2. & \quad 4A + 2B + C = 8 \\ 3. & \quad 9A + 3B + C = 15 \\ \end{aligned} \][/tex]
### Step 4: Solve the System of Equations
Subtract equation 1 from equation 2:
[tex]\[ (4A + 2B + C) - (A + B + C) = 8 - 3 \][/tex]
[tex]\[ 3A + B = 5 \quad \text{(Equation 4)} \][/tex]
Subtract equation 2 from equation 3:
[tex]\[ (9A + 3B + C) - (4A + 2B + C) = 15 - 8 \][/tex]
[tex]\[ 5A + B = 7 \quad \text{(Equation 5)} \][/tex]
Subtract equation 4 from equation 5:
[tex]\[ (5A + B) - (3A + B) = 7 - 5 \][/tex]
[tex]\[ 2A = 2 \][/tex]
[tex]\[ A = 1 \][/tex]
Substitute [tex]\(A = 1\)[/tex] back into equation 4:
[tex]\[ 3(1) + B = 5 \][/tex]
[tex]\[ 3 + B = 5 \][/tex]
[tex]\[ B = 2 \][/tex]
Finally, substitute [tex]\(A = 1\)[/tex] and [tex]\(B = 2\)[/tex] back into equation 1:
[tex]\[ 1 + 2 + C = 3 \][/tex]
[tex]\[ 3 + C = 3 \][/tex]
[tex]\[ C = 0 \][/tex]
### Step 5: Formulate the nth Term
With [tex]\(A = 1\)[/tex], [tex]\(B = 2\)[/tex], and [tex]\(C = 0\)[/tex], the nth term of the sequence is:
[tex]\[ a_n = n^2 + 2n \][/tex]
Thus, the nth term of the quadratic sequence [tex]\(3, 8, 15, 24, 35, \ldots\)[/tex] is:
[tex]\[ \boxed{a_n = n^2 + 2n} \][/tex]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.