AFCAT Numerical Ability · Sequence and Series

Arithmetic Progression: Terms, Counting Terms and Sums

A sequence that rises or falls by the same amount at every step: find any term, which term a number is, the progression from two of its terms, the sum of n terms, and the terms hidden in a sum formula.

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Why this matters

Almost every question here is one of two formulas, the nth term or the sum of n terms, read forwards or backwards. Marks go on counting: from 4 to 40 in steps of 4 there are 10 terms, not 9. The HARD questions give the sum and ask how many terms, which needs a quadratic, or give the sum as a formula in n and ask for a term.

Concept 1 of 6: The nth term of an AP

An arithmetic progression (AP) adds the same number, the common difference

d

, at every step. To reach the nth term from the first you take

n - 1

steps, not

n

Definition

The common difference is any term minus the one before it: $d = a_2 - a_1$ . It can be negative or a fraction.
The nth term is $a_n = a + (n - 1)d$ , where $a$ is the first term.
Any one of $a$ , $d$ , $n$ and $a_n$ can be found from the other three.
A falling AP has $d < 0$ : for $20, 17, 14, \dots$ the common difference is $-3$ .

nth term of an AP

a_n = a + (n - 1)d

Worked example

Find the 12th term of the AP

3, 8, 13, 18, \dots

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Sequence and Series · Arithmetic Progression

Find the

15^{th}

term of the AP:

-5, -2, 1, 4, ...

n minus 1 steps, not n

The 10th term of

3, 8, 13, \dots

3 + 9 \times 5 = 48

. Using

10 \times 5

gives 53, one step too far.

Subtract in order

d

is a later term minus the earlier one. For

20, 17, 14

it is

17 - 20 = -3

. Reading it as 3 sends the progression the wrong way.

Drill 6 more on the nth term of an ap

Concept 2 of 6: Which term is it, and how many terms are there

Run the nth-term formula backwards. Set

a + (n - 1)d

equal to the number and solve for

n

. A whole-number

n

means the number is a term; a fraction means it is not. The position of the last term is the number of terms.

Definition

Is $x$ a term? Solve $x = a + (n - 1)d$ . It is a term only if $n$ comes out a positive whole number.
Number of terms from $a$ to the last term $l$ : $n = \dfrac{l - a}{d} + 1$ .
Terms below a bound $B$ : solve $a + (n - 1)d < B$ and take the largest whole $n$ . Check that the next term goes past $B$ .
'How many two-digit multiples of 6' is the same count: the AP $12, 18, \dots, 96$ .

Number of terms

n = \dfrac{l - a}{d} + 1

Worked example

How many terms are there in the AP

6, 10, 14, \dots, 102

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Sequence and Series · Arithmetic Progression

How many terms of the AP

2, 7, 12, 17, ...

are less than

100

Add the 1

From 6 to 102 in steps of 4 there are 24 steps but 25 terms. Dividing the gap by

d

counts the steps, not the terms.

A fraction means no

If solving gives

n - 1 = 17.4

, the number is not in the AP at all. Do not round it to the nearest term.

Drill 5 more on which term is it, and how many terms are there

Concept 3 of 6: An AP from two of its terms

Two terms fix an AP. The gap between them is a whole number of common differences: from the 3rd term to the 8th is 5 steps. Divide the gap by the steps to get

d

, then step back to the first term.

Definition

$a_q - a_p = (q - p)d$ , so $d = \dfrac{a_q - a_p}{q - p}$ .
Then the first term is $a = a_p - (p - 1)d$ .
The first term counts as a term: $a_1$ and $a_8$ are 7 steps apart.

Difference from two terms

d = \dfrac{a_q - a_p}{q - p}

Worked example

The 4th term of an AP is 14 and the 10th term is 32. Find the first term and the 20th term.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Sequence and Series · Arithmetic Progression

The

5^{th}

term of an AP is

25

and

9^{th}

term is

49

. Find the first term.

Count the steps, not the positions

Between the 4th and 10th terms there are

10 - 4 = 6

steps. Dividing the gap by 7, or by 10, gives the wrong

d

Drill 3 more on an ap from two of its terms

Concept 4 of 6: Sum of the first n terms

Pair the first term with the last, the second with the second last, and so on: every pair has the same total. So the sum is the number of terms times the average of the first and last terms.

Definition

With the last term $l$ known: $S_n = \dfrac n2 (a + l)$ .
Without it: $S_n = \dfrac n2 [2a + (n - 1)d]$ .
If only the last term is given, find the number of terms first (previous concepts).
Given the sum and asked for $n$ : put the sum into the second form, clear the 2, and solve the quadratic in $n$ . Only a positive whole root counts.

Sum of n terms of an AP

S_n = \dfrac n2 (a + l) = \dfrac n2 [2a + (n - 1)d]

Worked example

Find the sum of the first 16 terms of the AP

3, 7, 11, \dots

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 4 · Sequence and Series · Arithmetic Progression

How many terms of the AP

1, 4, 7, 10, ...

will give sum

715

Two roots, one answer

3n^2 + n - 310 = 0

has roots 10 and

-\tfrac{31}{3}

. A number of terms is a positive whole number, so only 10 counts.

2a inside the bracket

The bracket in

\dfrac n2 [2a + (n - 1)d]

holds twice the first term. Writing

a

there undercounts every pair, and the answer lands on a distractor.

Drill 6 more on sum of the first n terms

Concept 5 of 6: From a sum formula to the nth term

When the sum of

n

terms is given as a formula in

n

, the nth term is what the nth term adds to the sum:

S_n - S_{n-1}

. The first term is simply

S_1

Definition

$a_n = S_n - S_{n-1}$ for $n \ge 2$ , and $a_1 = S_1$ .
For one particular term, skip the algebra: $a_5 = S_5 - S_4$ .
If $S_n = pn^2 + qn$ , the AP has $d = 2p$ and first term $p + q$ , so $a_n = 2pn + (q - p)$ .

nth term from the sum

a_n = S_n - S_{n-1}

Worked example

The sum of

n

terms of an AP is

2n^2 + 5n

. Find the nth term and the 10th term.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 5 · Sequence and Series · Arithmetic Progression

If the sum of

n

terms of an AP is

3n^2 + 2n

, find the

n^{th}

term.

S of n is not the nth term

Putting

n = 6

into the sum formula gives the sum of six terms, not the sixth term. Subtract

S_5

from it.

Drill 2 more on from a sum formula to the nth term

Concept 6 of 6: Symmetric terms and the middle term

In an AP the middle term is the average of the terms around it. So three unknown terms are best written

a - d, a, a + d

: their sum is

3a

, and

d

drops out.

Definition

Three numbers in AP: write $a - d, a, a + d$ . The sum is $3a$ and the product is $a(a^2 - d^2)$ .
$a, b, c$ are in AP exactly when $2b = a + c$ : the middle one is the arithmetic mean of the other two.
With an odd number $n$ of terms, the middle term is the average of all of them, so the sum is $n$ times the middle term.
Four numbers in AP: write $a - 3d, a - d, a + d, a + 3d$ (the common difference is $2d$ ).

Three terms in AP

a - d,\; a,\; a + d \qquad \text{sum} = 3a

Worked example

Three numbers in AP add up to 24 and their product is 480. Find them.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 6 · Sequence and Series · Arithmetic Progression

Three numbers in AP have sum

21

and product

315

. Find the numbers.

Centre on the middle term

With

a, a + d, a + 2d

the sum is

3a + 3d

and both unknowns stay. Writing

a - d, a, a + d

makes the sum give

a

at once.

Both signs of d

d^2 = 4

gives

d = 2

d = -2

. Both give the same three numbers in opposite order, so neither is lost.

Drill 1 more on symmetric terms and the middle term

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (6)

The nth term of an AP
nth term of an AP
$a_n = a + (n - 1)d$
Which term is it, and how many terms are there
Number of terms
$n = \dfrac{l - a}{d} + 1$
An AP from two of its terms
Difference from two terms
$d = \dfrac{a_q - a_p}{q - p}$
Sum of the first n terms
Sum of n terms of an AP
$S_n = \dfrac n2 (a + l) = \dfrac n2 [2a + (n - 1)d]$
From a sum formula to the nth term
nth term from the sum
$a_n = S_n - S_{n-1}$
Symmetric terms and the middle term
Three terms in AP
$a - d,\; a,\; a + d \qquad \text{sum} = 3a$

Watch out for (10)

n minus 1 steps, not n
→ The nth term of an AP
Subtract in order
→ The nth term of an AP
Add the 1
→ Which term is it, and how many terms are there
A fraction means no
→ Which term is it, and how many terms are there
Count the steps, not the positions
→ An AP from two of its terms
Two roots, one answer
→ Sum of the first n terms
2a inside the bracket
→ Sum of the first n terms
S of n is not the nth term
→ From a sum formula to the nth term
Centre on the middle term
→ Symmetric terms and the middle term
Both signs of d
→ Symmetric terms and the middle term

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