nth Terms of Linear Sequences

• A linear sequence is an ordered list of numbers where each term is generated by adding a fixed value, known as the common difference ($d$), to the preceding term.

• The position of a term is denoted by the variable $n$, where $n = 1$ represents the first term, $n = 2$ the second, and so on; $n$ must always be a positive integer.

• The nth term (often written as $u_n$ or $a_n$) is a general rule or formula that describes the relationship between the position $n$ and the value of the term at that position.

• Because the rate of change is constant, the relationship between $n$ and the term value is linear, similar to the algebraic equation of a straight line $y = mx + c$.

• The formula for a linear sequence is typically expressed in the form $u_n = dn + c$, where $d$ represents the common difference and $c$ is a constant.

• The common difference ($d$) acts as the gradient or slope of the sequence; it tells us how much the sequence increases or decreases for every step in $n$.

• The constant ($c$) represents the 'zeroth term' ($u_0$), which is the theoretical value the sequence would have if $n=0$.

• Alternatively, the formula can be written as $u_n = a + (n-1)d$, where $a$ is the first term ($u_1$). Both forms are mathematically equivalent but offer different conceptual starting points.

Step-by-Step: Finding the nth Term

• Step 1: Find the common difference ($d$). Subtract the first term from the second term ($u_2 - u_1$). Ensure this difference is consistent across other consecutive terms.
• Step 2: Multiply $n$ by $d$. This creates the 'times table' part of the sequence (e.g., if $d=5$, start with $5n$).
• Step 3: Find the adjustment constant ($c$). Compare the first term of your $dn$ sequence to the actual first term of the given sequence. The difference is your constant $c$.
• Step 4: Combine into a formula. Write the final expression as $u_n = dn + c$.

Verification

• Always test your formula by substituting $n=1$ and $n=2$ to see if it generates the correct first and second terms of the original sequence.

• Negative Differences: If the sequence is decreasing, $d$ must be negative. A common mistake is treating a decreasing sequence as having a positive $d$ and then subtracting later.

• Solving for n: If asked 'Is 100 a term in this sequence?', set your formula equal to 100 ($dn + c = 100$) and solve for $n$. If $n$ is not a whole number, the value is not in the sequence.

• Large Term Calculation: To find the 100th term, simply substitute $n=100$ into your derived formula. This is much faster and more accurate than continuing the pattern manually.

• Check the Zeroth Term: A quick way to find $c$ is to work backward from the first term by subtracting $d$ once ($c = u_1 - d$).