nth Terms of Linear Sequences

Linear sequences are discrete versions of linear functions ($y = mx + c$); the common difference $d$ is analogous to the gradient $m$, and the adjustment $c$ is analogous to the y-intercept.

The formula $u_n = dn + c$ works because multiplying the position $n$ by the common difference $d$ accounts for the cumulative growth from the start of the sequence.

The constant $c$ represents the 'zeroth term' ($n=0$), which is the value the sequence would have held immediately before the first term.

Mathematically, the formula can also be expressed as $u_n = a + (n-1)d$, where $a$ is the first term; this highlights that to reach the $n^{th}$ term, you add the difference $d$ exactly $n-1$ times to the starting value.

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 several pairs of terms.

Step 2: Create the $dn$ term: Multiply the common difference by $n$. For example, if the difference is $5$, the first part of your formula is $5n$.

Step 3: Calculate the Adjustment ($c$): Compare the value of $dn$ when $n=1$ to the actual first term of the sequence. The difference between them is your constant $c$.

Step 4: Combine and Verify: Write the full formula $u_n = dn + c$ and test it using $n=2$ or $n=3$ to ensure it generates the correct subsequent terms.

Term vs. Position: It is vital to distinguish between $n$ (the 'address' or position) and $u_n$ (the 'resident' or value at that address).

Increasing vs. Decreasing: An increasing sequence has a positive $d$, while a decreasing sequence has a negative $d$, which must be reflected in the formula (e.g., $-3n + 20$).

The 'Zeroth Term' Shortcut: To find the constant $c$ quickly, subtract the common difference $d$ from the first term ($a - d = c$).

Testing Membership: If asked if a number (e.g., $105$) is in a sequence, set the nth term formula equal to that number ($dn + c = 105$) and solve for $n$. If $n$ is a positive integer, the number is in the sequence.

Verification: Always check your final formula against at least two different terms in the sequence to avoid simple arithmetic errors in the adjustment constant.

Negative Differences: When the sequence goes down, the coefficient of $n$ MUST be negative. A common mistake is to find the difference is $4$ but forget the minus sign if the sequence is decreasing.

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

Incorrect Adjustment: Forgetting that the $dn$ part of the formula must be adjusted to match the first term, not just left as $dn$.

$n$ as a non-integer: Forgetting that $n$ represents a position in a list; if a calculation for $n$ results in a decimal, that specific value does not exist in the sequence.

Sign Errors: Miscalculating the common difference when dealing with negative numbers or sequences that cross zero.