SPI Mathematics · Sequence and Series

The nth Term of an AP

In an arithmetic progression each term is the one before it plus a fixed common difference d, so the nth term is a + (n − 1)d.

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Why this matters

Seventeen questions in the bank, 13 EASY and 4 MODERATE, and one is from the 2022 paper. Almost every one is a single use of a + (n − 1)d: find a term, find which term a number is, find d from two terms, or count back from the last term.

Concept 1 of 4: What makes a list an AP

A list of numbers is an arithmetic progression (AP) when you add the same number each time to get the next one. That fixed number is the common difference. It can be positive, negative, zero or a fraction.

Definition

An AP is a list $a_1, a_2, a_3, \ldots$ in which $a_2 - a_1 = a_3 - a_2 = \ldots$
The common difference is $d = a_2 - a_1$ : a term minus the one before it, never the other way round.
To test a list, find every gap. One unequal gap and it is not an AP.
A rule such as $a_n = 3n - 1$ gives the terms when you put $n = 1, 2, 3, \ldots$ A rule of the form $a_n = pn + q$ always gives an AP with $d = p$ .
Surds: write each term as a multiple of the same root first, as in $\sqrt{12} = 2\sqrt{3}$ and $\sqrt{27} = 3\sqrt{3}$ .

Common difference

d = a_2 - a_1 = a_3 - a_2

Worked example

Write the first four terms of the sequence

a_n = 4n - 3

and find its common difference.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Sequence and Series · The nth Term of an AP

Which of the following is an A.P.?

Subtracting the wrong way round

The common difference is a term minus the term before it. For 20, 17, 14 it is

17 - 20 = -3

, not 3. A list that falls has a negative common difference.

Judging surds by their look

\sqrt{2}, \sqrt{4}, \sqrt{6}

looks evenly spaced but is not an AP: the gaps are about 0.59 and 0.45. Rewrite each surd as a multiple of one root before deciding.

Squares of an AP are not an AP

2, 4, 6 is an AP, but their squares 4, 16, 36 have gaps 12 and 20. Squaring the terms breaks the equal steps.

Drill 4 more on what makes a list an ap

Concept 2 of 4: The nth term formula

To reach the nth term you start at the first term and take n minus 1 steps of size d. So the 10th term is the first term plus 9 common differences, not 10.

Definition

$a_n = a + (n - 1)d$ , where $a$ is the first term and $d$ the common difference.
Find a term: put in the value of $n$ .
Find which term a number is: set $a + (n - 1)d$ equal to it and solve for $n$ . If $n$ is not a whole number, the number is not a term.
Find $a$ or $d$ : each fact about a term gives one equation.
Count the terms from $a$ to a last term $l$ : $n = \dfrac{l - a}{d} + 1$ . Counting the multiples of 7 between 100 and 200 uses this.
Two APs with an equal term: write both nth terms and set them equal.

The nth term and the number of terms

a_n = a + (n - 1)d \qquad n = \frac{l - a}{d} + 1

athe first term
dthe common difference
lthe last term

Worked example

Find the 12th term of the AP 4, 9, 14, ...

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q21Easy

Example 2 · Sequence and Series · The nth Term of an AP

For what value of n, the *n*th term of an arithmetic progressions 63, 65, 67... and 3, 10, 17... are equal?

Using n instead of n minus 1

The 10th term of 3, 7, 11, ... is

3 + 9 \times 4 = 39

. Adding

10 \times 4

gives 43, one step too far.

Forgetting the +1 when counting terms

From 12 to 96 in steps of 12 there are

\frac{96 - 12}{12} + 1 = 8

terms, not 7. Dividing the distance counts the steps; the terms are one more.

Starting the count at the wrong number

To count the three-digit multiples of 8, start at 104 (the first multiple of 8 above 99), not at 100, and end at 992, the last one below 1000.

Drill 5 more on the nth term formula

Concept 3 of 4: The gap between two terms

Between the 9th and the 15th term there are 6 steps, so the two terms differ by 6 common differences. The first term cancels, so you do not need it.

Definition

$a_m - a_k = (m - k)d$ . The first term cancels.
If two terms are given, divide their difference by the number of steps between them to get $d$ .
Then put $d$ back into either term to find the first term.

Difference of two terms

a_m - a_k = (m - k)d

Worked example

The 15th term of an AP is 30 more than its 9th term. Find the common difference.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Sequence and Series · The nth Term of an AP

If the23

^{\text{rd}}

term of an AP exceeds its 16

^{\text{th}}

term by 21, then the common difference is:

Counting the terms instead of the steps

From the 9th term to the 15th there are 6 steps, not 7. The gap is

(m - k)d

, not

(m - k + 1)d

Losing the sign

a_{10} - a_{14}

-4d

, because the later term is subtracted. With

d = 3

it is

-12

Drill 3 more on the gap between two terms

Concept 4 of 4: Terms counted from the end

Read an AP backwards from its last term and you get a new AP whose common difference has the opposite sign. The 3rd term from the end is two steps back from the last term.

Definition

If the last term is $l$ , the nth term from the end is $l - (n - 1)d$ .
Or reverse the list: it starts at $l$ and its common difference is $-d$ .
If the AP has $N$ terms, the nth term from the end is the $(N - n + 1)$ th term from the start.

nth term from the end

l - (n - 1)d

Worked example

Find the 4th term from the end of the AP 3, 8, 13, ..., 98.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 4 · Sequence and Series · The nth Term of an AP

The

6^{\text{th}}

term from the end of the A.P. 5, 2, - 1, - 4, ..., - 31, is

Getting the direction wrong in a falling AP

For 50, 46, 42, ..., 2 the common difference is

-4

, so the terms just before 2 are 6, 10, 14. Going back from the end you subtract

d

, which here adds 4.

Taking n steps back instead of n minus 1

The 5th term from the end is 4 steps back from the last term, so it is

l - 4d

, not

l - 5d

Drill 1 more on terms counted from the end

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 (4)

What makes a list an AP
Common difference
$d = a_2 - a_1 = a_3 - a_2$
The nth term formula
The nth term and the number of terms
$a_n = a + (n - 1)d \qquad n = \frac{l - a}{d} + 1$
The gap between two terms
Difference of two terms
$a_m - a_k = (m - k)d$
Terms counted from the end
nth term from the end
$l - (n - 1)d$

Watch out for (10)

Subtracting the wrong way round
→ What makes a list an AP
Judging surds by their look
→ What makes a list an AP
Squares of an AP are not an AP
→ What makes a list an AP
Using n instead of n minus 1
→ The nth term formula
Forgetting the +1 when counting terms
→ The nth term formula
Starting the count at the wrong number
→ The nth term formula
Counting the terms instead of the steps
→ The gap between two terms
Losing the sign
→ The gap between two terms
Getting the direction wrong in a falling AP
→ Terms counted from the end
Taking n steps back instead of n minus 1
→ Terms counted from the end

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