Problem 28 - Algebra 2

Question

Find the indicated term of each arithmetic sequence. \(a_{1}=-4, d=-9, n=20\)

Step-by-Step Solution

Answer

The 20th term is -175.

Step 1: Understand the Problem

We need to find the 20th term of an arithmetic sequence where the first term \(a_1\) is \(-4\) and the common difference \(d\) is \(-9\).

Step 2: Recall the Formula for an Arithmetic Sequence

The formula to find the nth term of an arithmetic sequence is given by: \( a_n = a_1 + (n-1) \cdot d \).

Step 3: Substitute the Given Values into the Formula

Substitute \(a_1 = -4\), \(d = -9\), and \(n = 20\) into the formula: \[a_{20} = -4 + (20-1) \cdot (-9)\]

Step 4: Simplify the Equation

Simplify the expression inside the parentheses first: \[a_{20} = -4 + 19 \cdot (-9)\]Then calculate \(19 \cdot (-9) = -171\).

Step 5: Perform the Final Calculation

Substitute \(-171\) into the equation: \[a_{20} = -4 - 171\]This reduces to \(a_{20} = -175\).

Key Concepts

nth term formula

The formula for the nth term of an arithmetic sequence is a handy tool that helps you find any term in a sequence, not just the first or the second. It is represented as:\[a_{n} = a_{1} + (n-1) \cdot d\]where,

• \(a_{n}\) is the nth term you want to find,
• \(a_{1}\) is the first term of the sequence,
• \(n\) is the term number you are looking to find,
• \(d\) is the common difference between consecutive terms.

This formula leverages the fact that an arithmetic sequence increases or decreases by a constant amount—this is what the \((n-1) \cdot d\) section describes. If you know the first term and the common difference, you can figure out any term by plugging the numbers into this formula and simplifying it.

common difference

The common difference in an arithmetic sequence is the amount by which each term increases or decreases to get to the next term. You can find this number by subtracting any term from the next term in the sequence. For example, if you see a sequence like 3, 7, 11, 15, the common difference is:\[7 - 3 = 4\] and \(11 - 7 = 4\), which are both the same.

• In a sequence like 5, 10, 15, the common difference is 5.
• For another like 20, 15, 10, the difference would be -5, since each term decreases by 5.

Understanding this concept is crucial because the common difference allows you to use the nth term formula effectively. It can be a positive number (if the sequence is increasing), a negative number (if the sequence is decreasing), or zero (if the sequence is constant).

first term of sequence

The first term of a sequence, often denoted as \(a_{1}\), forms the foundation from which the entire sequence is built. It is the starting point or the first number in the list of an arithmetic sequence. Knowing the first term is essential because it is the baseline from which all subsequent terms are calculated using the common difference. If you have a sequence like \(-4, -13, -22\), then \(-4\) is the first term.

• The first term is crucial in finding any other term using the nth term formula.
• By applying the formula \(a_{n} = a_{1} + (n-1) \cdot d\), the first term \(a_{1}\) sets the initial value for the sequence.

Without knowing the first term, calculating any subsequent term using the nth term formula would be impossible, as you would lack the original reference point in the pattern.