Math Problem Statement

Write the formula for the nth term of the given arithmetic sequence

Write the formula for the nth term of the given arithmetic sequence.

Solution

The image shows two arithmetic sequences:

  1. Sequence 1: 1,4,7,10,13,1, 4, 7, 10, 13, \dots
  2. Sequence 2: 235,485,735,985,123,\frac{23}{5}, \frac{48}{5}, \frac{73}{5}, \frac{98}{5}, 123, \dots

Finding the nth term for both sequences:

1. Sequence 1: 1,4,7,10,13,1, 4, 7, 10, 13, \dots

This is an arithmetic sequence where the first term a1=1a_1 = 1 and the common difference d=3d = 3. The general formula for the nn-th term of an arithmetic sequence is:

an=a1+(n1)da_n = a_1 + (n - 1) d

Substituting the values: an=1+(n1)3=1+3n3=3n2a_n = 1 + (n - 1) \cdot 3 = 1 + 3n - 3 = 3n - 2

So, the nth term of the first sequence is: an=3n2a_n = 3n - 2

2. Sequence 2: 235,485,735,985,\frac{23}{5}, \frac{48}{5}, \frac{73}{5}, \frac{98}{5}, \dots

This is also an arithmetic sequence, where the first term a1=235a_1 = \frac{23}{5} and the common difference is: d=485235=255=5d = \frac{48}{5} - \frac{23}{5} = \frac{25}{5} = 5

Using the general formula again: an=a1+(n1)da_n = a_1 + (n - 1) d

Substituting the values: an=235+(n1)5=235+5n5a_n = \frac{23}{5} + (n - 1) \cdot 5 = \frac{23}{5} + 5n - 5

Simplifying: an=5n+235255=5n25a_n = 5n + \frac{23}{5} - \frac{25}{5} = 5n - \frac{2}{5}

So, the nth term of the second sequence is: an=5n25a_n = 5n - \frac{2}{5}


Would you like further details or have any questions?

Here are five related questions:

  1. How do we determine the common difference in an arithmetic sequence?
  2. What is the relationship between the first term and the common difference in a sequence?
  3. How can you find the sum of the first nn terms of an arithmetic sequence?
  4. What is the nth term formula for a geometric sequence?
  5. How can you extend an arithmetic sequence beyond the 10th term?

Tip: To identify an arithmetic sequence, look for a constant difference between consecutive terms.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Sequences
Algebra

Formulas

nth term of an arithmetic sequence: a_n = a_1 + (n - 1) d
Common difference: d = a_2 - a_1

Theorems

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Suitable Grade Level

Grades 8-10