Arithmetic progressions

Q: Can the common difference be negative or zero?

Yes. With a negative d the terms get smaller (10, 7, 4, …); with d = 0 every term is the same.

An arithmetic progression (AP) is a list of numbers in which each term is found by adding the same number, the common difference $d$ , to the term before it. For example 3, 7, 11, 15, … is an AP with first term $a = 3$ and $d = 4$ .

Formulas

The nth term

a_{n} = a + (n - 1) d

Sum of the first n terms

S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} (a + l)

Common difference

d = a_{2} - a_{1} = a_{3} - a_{2} = \dots

Worked examples

Example 1. Find the 20th term of the AP 3, 7, 11, 15, …

$a = 3$ and $d = 7 - 3 = 4$ .
$a_{20} = 3 + (20 - 1) \times 4 = 3 + 76 = 79$ .

Answer: 79

Example 2. Which term of the AP 5, 9, 13, … is 101?

$a = 5, d = 4$ . Set $5 + (n - 1) \times 4 = 101$ .
$(n - 1) \times 4 = 96$ , so $n - 1 = 24$ and $n = 25$ .

Answer: The 25th term

Example 3. Find the sum of the first 15 terms of 2, 5, 8, …

$a = 2, d = 3, n = 15$ .
$S_{15} = \frac{15}{2} [2 \times 2 + 14 \times 3] = \frac{15}{2} \times 46 = 345$ .

Answer: 345

Example 4. Find the sum of the first 100 natural numbers.

1, 2, 3, …, 100 is an AP with $a = 1$ and last term $l = 100$ .
$S_{100} = \frac{100}{2} (1 + 100) = 50 \times 101 = 5050$ .

Answer: 5050

Common mistakes

Using $n$ instead of $n - 1$ in the nth term formula. The first term already counts once, so only $n - 1$ steps of $d$ are added.
Finding $d$ the wrong way round. It is a later term minus the earlier one, so for 20, 17, 14, … it is $- 3$ , not $3$ .
Mixing up $a_{n}$ (one term) and $S_{n}$ (the total of the first $n$ terms).

Practice questions with answers

1. Find the 10th term of AP 3, 8, 13, 18, ….

Show answer

Answer: A) 48

a = 3, d = 5. a₁₀ = 3 + 9·5 = 48.

2. For an AP, the nth term is given by

a_{n} = a + (n - 1) d

. What does 'd' represent here?

AThe first term
BThe common difference
CThe number of terms
DThe last term

Show answer

Answer: B) The common difference

a_{n} = a + (n - 1) d

, 'a' is the first term and 'd' is the common difference (the fixed gap between consecutive terms).

3. The sum of the first n terms of an AP with first term a and common difference d is given by which formula?

A $S = \frac{n}{2} [2 a + (n - 1) d]$
B $S = \frac{n}{2} [a + (n - 1) d]$
C $S = n [2 a + (n - 1) d]$
D $S = \frac{n}{2}$ [ $2 a$ + nd]

Show answer

Answer: A)

S = \frac{n}{2} [2 a + (n - 1) d]

The standard formula for the sum of

n

terms of an AP is

S_{n} = \frac{n}{2}

[2 a + (n - 1) d]

4. The numbers 7,

x

and 15 are in AP. Find

x

Show answer

Answer: D) 11

For three terms in AP the middle one is the average of the other two:

x = (7 + 15) /2 = 11

. 8 is 15 − 7; 22 forgets to halve.

5. The first term of an AP of monthly rents is Rs 5000 and the 6th month rent is Rs 6500. What is the monthly increase?

A600
B300
C330
D270

Show answer

Answer: B) 300

a 6 - a = 5 d = 6500 - 5000 = 1500, so d = \frac{1500}{5} = R s 300

6. The first term of an AP is 3 and the sum of its first 9 terms is 171. Find its common difference.

Show answer

Answer: C) 4

S_{9} = \frac{9}{2} [2 (3) + 8 d] = 171

→

6 + 8 d = 38

→

8 d = 32

→

d = 4

. Check with the AP 3, 7, 11, …:

S_{9} = \frac{9}{2} (6 + 32) = 171

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Frequently asked questions

What is an arithmetic progression?

A list of numbers with the same difference between every pair of neighbouring terms, such as 4, 9, 14, 19, …

How do you check whether a list of numbers is an AP?

Subtract each term from the next one. If every difference is the same, it is an AP.

Can the common difference be negative or zero?

Yes. With a negative $d$ the terms get smaller (10, 7, 4, …); with $d = 0$ every term is the same.

When should I use S = n/2 (a + l)?

When you know the first and last terms. If you know the common difference instead, use $\frac{n}{2} [2 a + (n - 1) d]$ .

Arithmetic progressions

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

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