Math Problem Statement

how many terms are in the sequence 4,7,10,...61?

Solution

To determine the number of terms in the sequence 4,7,10,,614, 7, 10, \ldots, 61, we can recognize this as an arithmetic sequence where:

  • The first term a=4a = 4.
  • The common difference d=74=3d = 7 - 4 = 3.

The general formula for the nn-th term of an arithmetic sequence is:

an=a+(n1)da_n = a + (n - 1) \cdot d

We need to find nn when the last term an=61a_n = 61. Plugging in the values:

61=4+(n1)361 = 4 + (n - 1) \cdot 3

Simplify the equation:

614=(n1)361 - 4 = (n - 1) \cdot 3

57=(n1)357 = (n - 1) \cdot 3

Now, divide both sides by 3:

n1=573n - 1 = \frac{57}{3}

n1=19n - 1 = 19

So,

n=20n = 20

Therefore, there are 20 terms in the sequence.


Would you like a detailed explanation of any step? Here are some related questions:

  1. How do you find the common difference in an arithmetic sequence?
  2. What is the formula for the sum of an arithmetic sequence?
  3. How do you find the nn-th term of an arithmetic sequence?
  4. Can this method be used for sequences with different patterns?
  5. What changes if the sequence is geometric instead of arithmetic?

Tip: Always identify the common difference or ratio when dealing with sequences—it’s the key to finding missing terms or sums!

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Math Problem Analysis

Mathematical Concepts

Arithmetic Sequence
Sequences and Series

Formulas

nth term formula for arithmetic sequence: a_n = a + (n - 1) * d

Theorems

Arithmetic Sequence Formula

Suitable Grade Level

Grades 8-10