nth Terms of Linear Sequences

• Linear Sequence: A sequence of numbers where the change from one term to the next is always the same addition or subtraction.

• Common Difference ($d$): The fixed numerical value added to each term to produce the subsequent term; it represents the 'rate of change' of the sequence.

• Term Position ($n$): An integer representing the rank or index of a number in the sequence (e.g., $n=1$ for the first term, $n=2$ for the second).

• nth Term ($u_n$): A general expression or formula that defines the value of a term based solely on its position $n$.

• Linearity: Linear sequences are discrete versions of linear functions ($y = mx + c$); the common difference $d$ corresponds to the gradient $m$, and the sequence values fall on a straight line when graphed.

• Additive Structure: Every term $u_n$ can be viewed as the first term plus $(n-1)$ jumps of the common difference, leading to the logical derivation of the formula.

• Constant First Difference: The defining characteristic of a linear sequence is that the 'first difference' (the difference between terms) is constant; if the first difference varies, the sequence is non-linear.

The $dn + c$ Method

• Step 1: Find the Common Difference ($d$): Subtract the first term from the second term ($u_2 - u_1$) to identify the constant amount being added.

• Step 2: Multiply by $n$: Start the formula with $dn$; this accounts for the growth rate of the sequence.

• Step 3: Find the Adjustment Constant ($c$): Determine what must be added to or subtracted from $d(1)$ to reach the actual first term $u_1$. This is often calculated as $c = u_1 - d$.

• Step 4: Assemble the Formula: Combine the parts into the final expression $u_n = dn + c$.

• Step 5: Verification: Substitute $n=2$ or $n=3$ into your new formula to ensure it produces the correct corresponding term in the sequence.

• Term Value vs. Term Position: It is critical to distinguish between $n$ (the 'address' of the number) and $u_n$ (the 'house' at that address).

• Increasing vs. Decreasing: If $d > 0$, the sequence is ascending; if $d < 0$, the sequence is descending and the formula will feature a negative coefficient for $n$.

• The 'Term 0' Shortcut: To find the constant $c$ quickly, imagine a 'zeroth' term ($n=0$) by subtracting the common difference $d$ from the first term $u_1$.

• Testing Large Terms: If asked to find the 100th term, always find the $n$th term formula first rather than adding the difference repeatedly, which is prone to calculation errors.

• Reverse Engineering: If asked if a specific number (e.g., 505) is in a sequence, set the formula equal to that number ($dn + c = 505$) and solve for $n$; if $n$ is a positive integer, the number belongs to the sequence.

• Sanity Check: Always check the sign of your $d$. If the sequence is going down (e.g., $10, 7, 4...$), your formula MUST start with a negative $n$ term (e.g., $-3n$).

• Confusing $d$ and $c$: Students often use the first term as the coefficient for $n$ instead of the common difference.

• Off-by-one Errors: Using $u_n = a + nd$ instead of $u_n = a + (n-1)d$ results in skipping the first term's correct value.

• Negative Number Handling: When the common difference is negative, students frequently forget to treat the subtraction correctly when calculating the adjustment constant $c$.