Arithmetic Progression Explained — Sequence and Series (JAMB / WAEC Maths)
An arithmetic progression (AP) is a sequence of numbers in which each term after the first is found by adding a fixed number, called the common difference, to the previous term. This topic is a regular feature in JAMB and WAEC Mathematics, usually tested through finding a specific term or the sum of a number of terms. This lesson walks through both with clear worked examples.
Quick takeaways
- An arithmetic progression (AP) is a sequence where each term is obtained by adding a constant value called the common difference (d) to the previous term.
- The nth term of an AP is given by the formula: Tₙ = a + (n − 1)d, where “a” is the first term.
- The sum of the first n terms is given by: Sₙ = n/2 [2a + (n − 1)d], or equivalently Sₙ = n/2 (a + l), where “l” is the last term.
- Unlike a geometric progression, an arithmetic progression grows (or shrinks) by a constant amount at each step, not a constant ratio.
- Exam questions often give you two terms of an AP and ask you to find the first term and common difference before solving the actual question — practice this two-step setup.
What is an arithmetic progression?
An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed, constant value called the common difference (d) to the previous term.
For example, in the sequence 3, 7, 11, 15…, each term is obtained by adding 4 to the previous one, so the common difference is 4. This is different from a geometric progression, where each term is found by multiplying by a fixed ratio rather than adding a fixed amount.
How do you find the nth term of an arithmetic progression?
The nth term of an AP is calculated using the formula:
Tₙ = a + (n − 1)d
Where:
- a = the first term of the sequence
- d = the common difference
- n = the term number you want to find
For example, to find the 6th term of the sequence 3, 7, 11…, where a = 3 and d = 4: T₆ = 3 + (6 − 1)(4) = 3 + 20 = 23.
How do you find the sum of the first n terms of an arithmetic progression?
The sum of the first n terms of an AP can be calculated using either of these equivalent formulas:
Sₙ = n/2 [2a + (n − 1)d]
or, if the last term (l) is known:
Sₙ = n/2 (a + l)
The second formula is often quicker to use when the first and last terms are already known, since it avoids needing the common difference at all.
How do you solve AP questions when only two terms are given?
Many JAMB/WAEC questions give you two specific terms of an AP (not necessarily the first two) and ask you to find the first term, common difference, or a different specific term. The general approach:
- Write out the formula for each given term using Tₙ = a + (n − 1)d
- Subtract one equation from the other to eliminate “a” and solve for “d”
- Substitute the value of “d” back into either equation to find “a”
- Use these values to answer the actual question asked
How is an arithmetic progression different from a geometric progression?
The key distinction is how each term relates to the one before it:
- In an arithmetic progression, terms increase or decrease by a constant amount (addition/subtraction)
- In a geometric progression, terms increase or decrease by a constant ratio (multiplication/division)
This distinction is frequently tested directly, asking students to identify whether a given sequence is arithmetic or geometric before solving further.
Common mistakes students make with arithmetic progression
- Confusing the AP formulas with GP formulas, especially when a question doesn’t explicitly state which type of sequence is being used
- Forgetting to subtract 1 from n in the nth term formula (it’s (n − 1)d, not nd)
- Making sign errors when the common difference is negative (a decreasing sequence)
- Mixing up the two sum formulas and using the wrong one when the last term isn’t actually known
Frequently asked questions
Q: What is the formula for the nth term of an arithmetic progression? The nth term is given by Tₙ = a + (n − 1)d, where “a” is the first term, “d” is the common difference, and “n” is the term number.
Q: What is the difference between an arithmetic progression and a geometric progression? In an arithmetic progression, each term is found by adding a constant common difference to the previous term, while in a geometric progression, each term is found by multiplying the previous term by a constant common ratio.
Q: What are the two formulas for the sum of an arithmetic progression, and when should each be used? Sₙ = n/2 [2a + (n − 1)d] is used when you know the first term and common difference, while Sₙ = n/2 (a + l) is quicker to use when both the first and last terms are already known.
Q: How do you find the common difference if you’re only given two non-consecutive terms? Write the formula for each given term using Tₙ = a + (n − 1)d, then subtract one equation from the other to eliminate “a” and solve directly for the common difference “d”.
New to sequences and series? Compare this with Geometric Progression →
This is one topic from our full JAMB & WAEC Maths course — see everything covered →
