Math Problem Statement
Solution
The sequence provided is a quadratic sequence: 7, 10, 15, 22, 31, ...
Step 1: Identify the first and second differences
Let's calculate the first and second differences:
-
First differences:
-
Second differences:
Since the second differences are constant and equal to 2, we confirm that this is a quadratic sequence.
Step 2: General form of a quadratic sequence
The general form of a quadratic sequence is:
Where , , and are constants to be determined. We now have to solve for , , and using the given terms of the sequence.
Step 3: Set up the system of equations
Using the first three terms of the sequence:
-
For , :
-
For , :
-
For , :
Step 4: Solve the system of equations
We now solve the system of equations:
By solving these equations step-by-step, we will find the values of , , and .
Let's solve it.The solution to the system of equations is:
Thus, the th term rule for the quadratic sequence is:
Step 5: Find the position of the term with value 150
Now, to find the position of the term with value 150, we set :
Solving for :
Therefore, the term with value 150 is the 12th term in the sequence.
Would you like further clarification on any part of this solution?
Here are 5 related questions for further practice:
- How would the formula change if the sequence started with a different number?
- Can you determine the 15th term in this sequence using the formula?
- What are the differences between a linear and quadratic sequence in terms of their nth term formulas?
- How would you find the position of a different number, such as 200, in this sequence?
- Can you derive the nth term formula for a cubic sequence?
Tip: For any quadratic sequence, the second differences between terms are always constant, which is the key to identifying and solving for the nth term.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Quadratic Sequences
Algebra
Patterns
Formulas
nth term of quadratic sequence: an^2 + bn + c
Solving system of equations for a, b, c
Square root to solve quadratic equation
Theorems
General form of a quadratic sequence
Solving simultaneous equations
Using the square root to solve for n
Suitable Grade Level
Grades 9-11
Related Recommendation
Solving the 10th Term of a Quadratic Sequence 7, 14, 25, 40, 59
Finding the nth Term of a Quadratic Sequence: 1, 3, 7, 13, 21
Find the nth term of the sequence: 50, −10, 2
Find the General Term of a Quadratic Sequence and the 1839th Term
Find the 100th Term of a Quadratic Sequence Using Un = an² + bn + c