Math Problem Statement

The start of a quadratic sequence is shown below. By first finding the nth term rule, work out the position of the term with value 150 in the sequence: 7, 10, 15, 22, 31, ...

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: 107=310 - 7 = 3 1510=515 - 10 = 5 2215=722 - 15 = 7 3122=931 - 22 = 9

  • Second differences: 53=25 - 3 = 2 75=27 - 5 = 2 97=29 - 7 = 2

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:

an=an2+bn+ca_n = an^2 + bn + c

Where aa, bb, and cc are constants to be determined. We now have to solve for aa, bb, and cc using the given terms of the sequence.

Step 3: Set up the system of equations

Using the first three terms of the sequence:

  • For n=1n = 1, a1=7a_1 = 7: a(1)2+b(1)+c=7a+b+c=7a(1)^2 + b(1) + c = 7 \quad \Rightarrow \quad a + b + c = 7

  • For n=2n = 2, a2=10a_2 = 10: a(2)2+b(2)+c=104a+2b+c=10a(2)^2 + b(2) + c = 10 \quad \Rightarrow \quad 4a + 2b + c = 10

  • For n=3n = 3, a3=15a_3 = 15: a(3)2+b(3)+c=159a+3b+c=15a(3)^2 + b(3) + c = 15 \quad \Rightarrow \quad 9a + 3b + c = 15

Step 4: Solve the system of equations

We now solve the system of equations:

  1. a+b+c=7a + b + c = 7
  2. 4a+2b+c=104a + 2b + c = 10
  3. 9a+3b+c=159a + 3b + c = 15

By solving these equations step-by-step, we will find the values of aa, bb, and cc.

Let's solve it.The solution to the system of equations is:

a=1,b=0,c=6a = 1, \quad b = 0, \quad c = 6

Thus, the nnth term rule for the quadratic sequence is:

an=n2+6a_n = n^2 + 6

Step 5: Find the position of the term with value 150

Now, to find the position of the term with value 150, we set an=150a_n = 150:

n2+6=150n^2 + 6 = 150

Solving for nn:

n2=1506=144n^2 = 150 - 6 = 144

n=144=12n = \sqrt{144} = 12

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:

  1. How would the formula change if the sequence started with a different number?
  2. Can you determine the 15th term in this sequence using the formula?
  3. What are the differences between a linear and quadratic sequence in terms of their nth term formulas?
  4. How would you find the position of a different number, such as 200, in this sequence?
  5. 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.

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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