Harder arithmetic progressions

In General Maths you met the two A.P. formulas (see sequences and series↺): the $n$th term and the sum of $n$ terms.

Further Maths questions give you less directly: two facts to combine, a sum that leads to a quadratic in $n$, or a story to turn into an A.P. The method is always the same. Write every fact as an equation in $a$ and $d$ (or in $n$), then solve.

Two facts, two unknowns

Each piece of information becomes one equation. A term gives $a + (n - 1)d = \ldots$, and a sum gives $\frac n2[2a + (n - 1)d] = \ldots$. Simplify each, then solve them simultaneously.

How many terms? A quadratic in n

When the sum is given and $n$ is unknown, the sum formula becomes a quadratic in $n$. Solve it, and keep only the positive whole-number answer.

The least number of terms to pass a total

“The least number of terms for the sum to be greater than 250” is an inequality, $S_n > 250$. Solve the matching equation, then round the root up to the next whole number: every smaller $n$ falls short.

Slide $n$ and watch the columns cross the target:

Word problems

A regular increase each week or month is an A.P. Decide what $a$, $d$ and $n$ are before using a formula, and whether the question wants one payment ($T_n$) or the total so far ($S_n$).

For example, a trader saves ₦500 in the first month and ₦150 more each month after that. The total saved in a year:

• ${a = 500}$, ${d = 150}$, ${n = 12}$, and the question wants the total, $S_{12}$.
• ${S_{12} = \frac{12}{2}[2(500) + 11(150)]}$.
• ${= 6(1000 + 1650) = 6 \times 2650}$.
• So the trader saves ₦15 900.

Sequences defined by a rule

Some sequences give each term from the ones before it, not from $n$. Start from the terms you are given and build the next one, one step at a time.

A sequence can also be given by a formula for its sum, $S_n$. Then each term is the difference of two sums, $T_n = S_n - S_{n - 1}$ (see sequences and series↺).

Your turn