Math Problem Statement

1 7 19 43 91 find the explicit formula

Solution

The given sequence is: 1, 7, 19, 43, 91.

Step 1: Identify the pattern

This is a sequence that seems to grow more rapidly with each step. To check, let's find the differences between consecutive terms:

19 - 7 = 12 \\ 43 - 19 = 24 \\ 91 - 43 = 48$$ ### Step 2: Check the second differences Now, let's find the second differences (differences between the first differences): $$12 - 6 = 6 \\ 24 - 12 = 12 \\ 48 - 24 = 24$$ The second differences are not constant, but let's check the third differences: $$12 - 6 = 6 \\ 24 - 12 = 12$$ The **third differences** are constant, which suggests this is a cubic sequence. Therefore, the explicit formula is likely of the form: $$a_n = an^3 + bn^2 + cn + d$$ ### Step 3: Solve for $$a$$, $$b$$, $$c$$, and $$d$$ We will now use the given terms to set up a system of equations by plugging in values of $$n$$ for each term in the sequence: 1. For $$n = 1$$, $$a_1 = 1$$: $$a(1^3) + b(1^2) + c(1) + d = 1 \quad \Rightarrow \quad a + b + c + d = 1$$ 2. For $$n = 2$$, $$a_2 = 7$$: $$a(2^3) + b(2^2) + c(2) + d = 7 \quad \Rightarrow \quad 8a + 4b + 2c + d = 7$$ 3. For $$n = 3$$, $$a_3 = 19$$: $$a(3^3) + b(3^2) + c(3) + d = 19 \quad \Rightarrow \quad 27a + 9b + 3c + d = 19$$ 4. For $$n = 4$$, $$a_4 = 43$$: $$a(4^3) + b(4^2) + c(4) + d = 43 \quad \Rightarrow \quad 64a + 16b + 4c + d = 43$$ ### Step 4: Solve the system of equations Now, solve the system of four equations: 1. $$a + b + c + d = 1$$ 2. $$8a + 4b + 2c + d = 7$$ 3. $$27a + 9b + 3c + d = 19$$ 4. $$64a + 16b + 4c + d = 43$$ By solving this system, we get: $$a = 2, \quad b = -3, \quad c = 0, \quad d = 2$$ ### Step 5: Write the explicit formula Thus, the explicit formula for the sequence is: $$a_n = 2n^3 - 3n^2 + 2$$ This is the explicit formula for the given sequence. Would you like further details or have any questions? --- Here are some related questions you might find helpful: 1. What is the next term in the sequence using the formula $$a_n = 2n^3 - 3n^2 + 2$$? 2. Can you find an explicit formula for a quadratic sequence? 3. How do you verify the explicit formula by plugging in values for $$n$$? 4. What are the differences between linear, quadratic, and cubic sequences? 5. How can you determine the degree of a sequence? **Tip:** Recognizing the pattern of differences helps in identifying the degree of the polynomial formula for the sequence.

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Math Problem Analysis

Mathematical Concepts

Sequences
Cubic Sequences
Polynomial Functions

Formulas

General cubic formula: a_n = an^3 + bn^2 + cn + d
Difference method to identify sequence patterns

Theorems

Polynomial degree identification through differences

Suitable Grade Level

Grades 10-12

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