Question

write a rule for the nth term of the arithmetic sequence. \(a_{12}=-38, a_{19}=-73\)

Short answer

The rule for the nth term of the given arithmetic sequence is \(a_n = 17 + (n-1)*-5\).

Step-by-step solution

Identify given values

The terms given to us are the 12th term and the 19th term of the arithmetic sequence, they are \(a_{12} = -38\) and \(a_{19} = -73\) respectively.

Find the common difference

To find 'd', the common difference, you subtract the 12th term from the 19th term and then divide by the difference in their positions (19-12).\nSo, \(d = (a_{19} - a_{12}) / (19-12)\) => \(d = (-73 - -38) / 7 = -5\).

Find the first term

Rearrange the formula for the nth term of an arithmetic sequence to find the first term 'a'. Substituting \(a_{12}\) and 'd' into the formula we get: \(a_{12} = a + 11*d\) => \(-38 = a + 11*(-5)\).\nSolve for 'a' to get: \(a = -38 -11*(-5) = -38+55 = 17\).

Write the rule for the nth term

The nth term, \(a_n\), of an arithmetic sequence can be presented as \(a_n = a + (n-1) * d\). Substitute 'a' and 'd' into this equation to get the rule for the nth term.\nThe nth term of the objective arithmetic sequence is hence given by \(a_n = 17 + (n-1)*-5\).

Key concepts

Common Difference

In an arithmetic sequence, the common difference is a crucial concept that defines the pattern of the sequence. This difference, often denoted by the letter 'd', is the fixed amount by which each term in the sequence increases or decreases from the previous one.
To find the common difference, you simply subtract any term from the term that follows it. In the context of the example provided, you would subtract the 12th term (\(a_{12} = -38\) ) from the 19th term (\(a_{19} = -73\) ).
Here's how it works in this scenario:

• Calculate the difference between the term numbers: 19 - 12 = 7.
• Subtract the 12th term from the 19th term: \(-73 - (-38) = -73 + 38 = -35\).
• Divide the result by the difference between their positions: \(-35 / 7 = -5\).

Thus, the common difference here is -5, indicating that each term in the sequence is 5 less than the term before it.

Nth Term Formula

The Nth Term Formula is a handy tool for determining any term in an arithmetic sequence without having to list all the preceding terms. The formula is expressed as \(a_n = a + (n-1) \cdot d\), where:

• \(a_n\) is the nth term you want to find.
• \(a\) is the first term of the sequence.
• \(d\) is the common difference between the terms.
• \(n\) is the position of the term you want to find.

To use this formula, you replace the variables with their respective values. For instance:

• Find 'a': Using \(a_{12} = -38\), \(-38 = a + 11 \cdot (-5)\) simplifies to \(a = 17\).
• Now, plug 'a' and 'd' into the formula to get the nth term: \(a_n = 17 + (n-1) \cdot (-5)\).

This formula is instrumental in saving time and making it easy to find any term in the sequence quickly.

Arithmetic Sequence Rule

The Arithmetic Sequence Rule is a fundamental principle that dictates how terms are generated in an arithmetic sequence. It establishes that the progression from one term to the next results from adding a constant value, known as the common difference, to the previous term.
Here's a simplified understanding:

• Start from the first term, 'a'.
• Add the common difference 'd' repeatedly to get the subsequent terms.
• Continue this process to reach any desired term in the sequence.

Using the formula \(a_n = a + (n-1) \cdot d\), each term is easily computed. For example, in the given exercise, where the first term \(a = 17\) and the common difference \(d = -5\), every term in this sequence decreases by 5 as you progress from one term to the next.
This systematic approach allows for a predictable and orderly calculation of each sequence term, reinforcing the nature of arithmetic sequences as linear, evenly spaced sets of numbers.