TheIBAxolotl
The idea
An arithmetic sequence changes by the same amount every step. Seats in rows that each have 3 more than the row before, a savings jar that gets USD 20 more every week, a taxi fare that rises by a fixed amount per kilometre: all of these are arithmetic. The fixed step is the common difference \(d\); it can be negative for a decreasing sequence.
The nth term
To reach term \(n\) you start at \(u_1\) and take \(n-1\) steps of size \(d\):
\[u_n = u_1 + (n-1)d\]
Two pieces of information (for example two terms) are enough to find \(u_1\) and \(d\): write two equations and solve them simultaneously, by hand or with the GDC's equation solver.
The sum of n terms
Pair the first term with the last, the second with the second-last, and so on: every pair has the same total. That gives
\[S_n = \frac{n}{2}(u_1 + u_n) = \frac{n}{2}\big(2u_1 + (n-1)d\big)\]
Use the first version when you know the last term, the second when you know \(d\).
Sigma notation
\(\displaystyle\sum_{k=1}^{10}(3k+2)\) means "substitute \(k = 1, 2, \dots, 10\) and add". The terms are \(5, 8, \dots, 32\), so the sum is \(\frac{10}{2}(5+32) = 185\). A GDC can evaluate sigma expressions directly, which is a good check.
Worked example
An outdoor theatre has 15 rows. The front row has 20 seats and each row has 3 more seats than the row in front of it.
- Last row: \(u_{15} = 20 + 14 \times 3 = 62\) seats.
- Total seats: \(S_{15} = \frac{15}{2}(20 + 62) = 615\).
- Which row first has more than 50 seats? Solve \(20 + 3(n-1) > 50\), so \(n - 1 > 10\), so \(n > 11\): the 12th row (which has 53).
Simple interest and linear models
Simple interest adds the same amount every year, so the balance forms an arithmetic sequence. With capital \(C\) and annual rate \(r\%\), the interest after \(n\) years is \(I = \dfrac{C r n}{100}\). Real data that rise by roughly the same amount each step can be modelled as arithmetic: estimate \(d\) from the average step, then say clearly that the model is an approximation.
GDC tips
- Use the sequence or table mode to list terms and find the first term that passes a target.
- The numeric solver handles $S_n = $ target when \(n\) is unknown; remember \(n\) must be a positive integer and round in the direction the context needs.
Common mistakes
- Using \(n d\) instead of \((n-1)d\) in the nth term.
- Confusing a term (one row) with a sum (all rows so far).
- Giving a non-integer \(n\), or rounding down when the question asks when a target is first exceeded.
- Forgetting that a decrease means \(d\) is negative.
Exam tips
- Write the formula with numbers substituted before using the calculator: it earns method marks even if you mistype.
- Read "in the 8th week" versus "after 8 weeks" carefully; they can differ by one term.
- State units and interpret the answer (rows, seats, dollars).