find the general form of the series and find their sum.
1+9+45+81

To find the general form of the given series and their sum, let's break down the steps:

### Step 1: Observe the Sequence
The sequence provided is: 
\[ 1, 9, 45, 81 \]

### Step 2: Identify a Pattern
Look for a pattern in the differences between consecutive terms, or try to express the sequence as a function of \( n \).

- First term (\(a_1\)) = 1
- Second term (\(a_2\)) = 9
- Third term (\(a_3\)) = 45
- Fourth term (\(a_4\)) = 81

This looks like a quadratic sequence, meaning we can express the general term \( a_n \) in the form:
\[
a_n = an^2 + bn + c
\]
Let's calculate the differences:

1. First difference:
   - \( 9 - 1 = 8 \)
   - \( 45 - 9 = 36 \)
   - \( 81 - 45 = 36 \)

2. Second difference:
   - \( 36 - 8 = 28 \)
   - \( 36 - 36 = 0 \)

The constant second difference indicates a quadratic relationship.

### Step 3: Set Up a Quadratic Formula
Assume the general form is \( a_n = an^2 + bn + c \). Now, let's solve for \( a \), \( b \), and \( c \) using the known terms:
- \( a_1 = 1 \)
- \( a_2 = 9 \)
- \( a_3 = 45 \)

#### System of Equations
For \( n = 1 \):
\[
a(1)^2 + b(1) + c = 1 \quad \Rightarrow \quad a + b + c = 1
\]
For \( n = 2 \):
\[
a(2)^2 + b(2) + c = 9 \quad \Rightarrow \quad 4a + 2b + c = 9
\]
For \( n = 3 \):
\[
a(3)^2 + b(3) + c = 45 \quad \Rightarrow \quad 9a + 3b + c = 45
\]

Now solve this system of equations:
1. \( a + b + c = 1 \)
2. \( 4a + 2b + c = 9 \)
3. \( 9a + 3b + c = 45 \)

### Step 4: Solve the System of Equations
We will solve this system to find the coefficients \( a \), \( b \), and \( c \).from sympy import symbols, Eq, solve

# Define the variables
a, b, c = symbols('a b c')

# Define the system of equations based on the known terms
eq1 = Eq(a + b + c, 1)     # Equation for a_1 = 1
eq2 = Eq(4*a + 2*b + c, 9)  # Equation for a_2 = 9
eq3 = Eq(9*a + 3*b + c, 45) # Equation for a_3 = 45

# Solve the system of equations
solution = solve([eq1, eq2, eq3], (a, b, c))
solutionThe solution to the system of equations is:
- \( a = 14 \)
- \( b = -34 \)
- \( c = 21 \)

Thus, the general form of the series is:
\[
a_n = 14n^2 - 34n + 21
\]

### Step 5: Find the Sum of the Series
Now, to find the sum of the series up to the 4th term, substitute \( n = 1 \), \( n = 2 \), \( n = 3 \), and \( n = 4 \) into the general form and sum them up.

Let me calculate the sum for the first four terms.# Define the general term function for the series
def general_term(n):
    return 14*n**2 - 34*n + 21

# Calculate the sum of the first four terms
sum_series = sum(general_term(n) for n in range(1, 5))
sum_seriesThe sum of the series for the first four terms is:
\[
1 + 9 + 45 + 81 = 164
\]

### Summary:
- The general form of the series is: \( a_n = 14n^2 - 34n + 21 \).
- The sum of the first four terms is 164.

Do you have any questions or would you like more details on any of the steps?

Here are 5 related questions for further practice:
1. Find the next term in the series after 81.
2. What is the sum of the first 6 terms of the series?
3. How does the general form change if the sequence starts with a different first term?
4. Can you derive a different quadratic sequence by modifying the constants \( a \), \( b \), and \( c \)?
5. How do you determine if a sequence is quadratic based solely on the differences?

**Tip:** When working with sequences, always check if a pattern exists in the differences between terms. This can help identify whether the sequence is linear, quadratic, or of a higher order.

Explore how to find the general form and sum of the quadratic sequence 1, 9, 45, 81. Learn about quadratic sequences, solving systems of equations, and calculating summations. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation