A-Level Maths 9709 / Topic 1.6 / AS & A Level

Add the same amount, every time.

An arithmetic progression grows by a fixed step, the common difference. Two formulae, one for the $n$th term and one for the sum of $n$ terms, answer every question about it, from savings plans to the sum of all multiples of 3 between 100 and 300.

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The key idea

An AP with first term $a$ and common difference $d$ has $n$th term $u_n = a + (n-1)d$ and sum of the first $n$ terms $S_n = \dfrac{n}{2}\left[2a + (n-1)d\right] = \dfrac{n}{2}(a + l)$, where $l$ is the last term. Three numbers $a, b, c$ are in AP when $2b = a + c$.

Fig. 1: the AP 2, 5, 8, 11, 14 (solid) and the same terms reversed (dashed) fit together into a rectangle 5 columns wide and a + l = 16 high, so twice the sum is 5 × 16 and the sum is 40

Section 01

The common difference.

In the lab, set the first term and the common difference with the sliders and watch each term grow by the same step. Notice that to reach term $n$ you take only $n - 1$ steps from the first term: that is where the $n - 1$ in the formula comes from.

AP: each term is the previous term plus a constant $d$, the common difference (which may be negative or a fraction).
$n$th term: $u_n = a + (n-1)d$.
Three terms in AP: $b - a = c - b$, i.e. $2b = a + c$ (the middle term is the mean of its neighbours).

Worked example A term, and which term

For the AP $7, 11, 15, \ldots$ find the 20th term, and find which term is equal to 203.

$a = 7$, $d = 4$. $u_{20} = 7 + 19 \times 4 = 83$.
Set $7 + (n-1) \times 4 = 203$: $(n-1) \times 4 = 196$, so $n - 1 = 49$.

Answer: $u_{20} = 83$; 203 is the 50th term

Worked example Two terms give a and d

The 5th term of an AP is 18 and the 12th term is 46. Find the first term and the common difference.

$a + 4d = 18$ and $a + 11d = 46$.
Subtract: $7d = 28$, so $d = 4$.
Then $a = 18 - 16 = 2$.

Answer: $a = 2$, $d = 4$

Worked example Using 2b = a + c

The numbers $x$, $2x + 1$ and $5x - 4$ are consecutive terms of an AP. Find $x$ and the three terms.

$2(2x + 1) = x + (5x - 4)$.
$4x + 2 = 6x - 4$, so $2x = 6$ and $x = 3$.
The terms are $3, 7, 11$, which do go up by 4 each time.

Answer: $x = 3$; the terms are 3, 7, 11

Examiner trap

The 20th term is $a + 19d$, not $a + 20d$: the first term already counts as one of the twenty.

Section 02

Summing an AP.

Write the sum forwards and backwards and add the two lines, as in Fig. 1. Every pair adds to $a + l$, and there are $n$ pairs, so $2S_n = n(a + l)$. Replacing $l$ by $a + (n-1)d$ gives the second form.

$S_n = \dfrac{n}{2}(a + l)$ when you know the last term.
$S_n = \dfrac{n}{2}\left[2a + (n-1)d\right]$ when you know $d$.
Any single term: $u_n = S_n - S_{n-1}$, and $u_1 = S_1$.

Worked example A straight sum

Find the sum of the first 30 terms of the AP $7, 11, 15, \ldots$

$a = 7$, $d = 4$, $n = 30$.
$S_{30} = \dfrac{30}{2}\left[14 + 29 \times 4\right] = 15 \times 130$.

Answer: $1950$

Worked example Multiples between two numbers

Find the sum of all the multiples of 3 between 100 and 300.

The first multiple above 100 is 102; the last below 300 is 297.
Number of terms: $\dfrac{297 - 102}{3} + 1 = 65 + 1 = 66$.
$S = \dfrac{66}{2}(102 + 297) = 33 \times 399$.

Answer: $13\,167$

Worked example How many terms?

The AP $5, 8, 11, \ldots$ is summed term by term. Find the least number of terms for the sum to exceed 1000.

$S_n = \dfrac{n}{2}\left[10 + 3(n-1)\right] = \dfrac{n(3n + 7)}{2}$.
Solve $\dfrac{n(3n + 7)}{2} > 1000$, i.e. $3n^2 + 7n - 2000 > 0$. The positive root is $n = \dfrac{-7 + \sqrt{24\,049}}{6} = 24.68$.
So $n$ must be at least 25. Check: $S_{24} = 948$ and $S_{25} = 1025$.

Answer: 25 terms

Your turn

The sum of the first $n$ terms of a progression is $S_n = 3n^2 + 2n$. Find the first term and the common difference.

Examiner trap

Counting terms between two numbers: $\dfrac{297 - 102}{3} = 65$ counts the gaps; there is one more term than gaps, so $n = 66$.

Stage 1 · Learn

Check your understanding

Four quick checks. Each correct answer earns XP and records this skill on your profile.

Quick check+10 XP

Find the 10th term of the AP $3, 8, 13, \ldots$

Quick check+10 XP

What is the common difference of the AP $20, 17, 14, \ldots$?

Quick check+10 XP

Find the sum of the first 20 positive integers.

Quick check+10 XP

The numbers $a$, $b$, $c$ are consecutive terms of an AP. Which must be true?

Section 03

Problems in context.

Wages that rise by a fixed amount, savings that grow by a fixed sum, rows of seats that increase by two: each is an AP. Decide what $a$, $d$ and $n$ stand for, and whether the question wants one term (how much in month 24) or a sum (how much altogether).

Worked example A savings plan

Ana saves $50 in the first month and increases her saving by $5 each month. (a) How much does she save in total in the first two years? (b) In which month does her monthly saving first exceed $150?

(a) Two years is 24 months: $a = 50$, $d = 5$, $n = 24$.
$S_{24} = 12\left[100 + 23 \times 5\right] = 12 \times 215 = 2580$.
(b) Monthly saving $u_n = 50 + 5(n-1) > 150$, so $5(n-1) > 100$, $n - 1 > 20$, $n > 21$.
Month 21 is exactly $150, so the first month that exceeds it is month 22.

Answer: (a) $2580 (b) month 22

Examiner trap

Read whether the question asks for one term or the total. 'How much in the 24th month' is $u_{24}$; 'how much in the first 24 months' is $S_{24}$.

Stage 2 · Exam

Exam-style questions

Unlocks once the four checks above are done. Worth more XP, written to AS & A Level Paper 1 to 6 standard.

Finish the four checks above to unlock the exam questions

Exam style+20 XP

The 3rd term of an AP is 11 and the 8th term is 31. Find the sum of the first 20 terms.

Exam style+20 XP

Find the sum of all the multiples of 7 between 100 and 500.

Exam style+20 XP

The sum of the first $n$ terms of an AP is $S_n = 2n^2 + 5n$. Find the 10th term.

Exam style+20 XP

Find the least number of terms of the AP $4, 7, 10, \ldots$ that must be taken for the sum to exceed 500.

Skill unlocked

Arithmetic progressions, summed up.

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Six original Cambridge-style questions

Terms and sums of APs, the condition $2b = a + c$, the proof of the sum formula, and a training plan.