Math Problem Statement

Write a recursive formula and an explicit formula for the following arithmetic sequence. -2​, 6​, 14​, 22​, 30​, ...

Solution

For the given arithmetic sequence 2,6,14,22,30,-2, 6, 14, 22, 30, \dots, let's determine the recursive and explicit formulas.

1. Recursive Formula

In an arithmetic sequence, each term is obtained by adding a common difference dd to the previous term. To find dd, we can subtract the first term from the second term: d=6(2)=8d = 6 - (-2) = 8

So, the recursive formula is: an=an1+8,a1=2a_n = a_{n-1} + 8, \quad a_1 = -2 where ana_n is the nn-th term, and a1a_1 is the first term of the sequence.

2. Explicit Formula

The explicit formula for an arithmetic sequence is given by: an=a1+(n1)da_n = a_1 + (n - 1)d

Substituting a1=2a_1 = -2 and d=8d = 8: an=2+(n1)8a_n = -2 + (n - 1) \cdot 8 Simplify the expression: an=2+8n8=8n10a_n = -2 + 8n - 8 = 8n - 10

So, the explicit formula for the sequence is: an=8n10a_n = 8n - 10

Final Formulas:

  • Recursive Formula: an=an1+8a_n = a_{n-1} + 8, with a1=2a_1 = -2.
  • Explicit Formula: an=8n10a_n = 8n - 10.

Would you like further details or explanations on these formulas?


Here are some related questions for further practice:

  1. How would you find the 10th term using the explicit formula?
  2. What is the common difference, and why is it essential for an arithmetic sequence?
  3. How do you convert a recursive formula to an explicit formula?
  4. Can you identify the type of sequence given only the recursive formula?
  5. How would the formulas change if the common difference was negative?

Tip: For arithmetic sequences, always identify the common difference first; it simplifies finding both recursive and explicit formulas.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Sequences
Recursive Formulas
Explicit Formulas

Formulas

Recursive formula: a_n = a_{n-1} + d
Explicit formula: a_n = a_1 + (n - 1) * d

Theorems

Properties of Arithmetic Sequences

Suitable Grade Level

Grades 9-10