What this chapter is about
An arithmetic progression (AP) is a sequence of numbers where each term after the first is obtained by adding a fixed number to the previous term. This fixed number is called the common difference. Examples surround us: monthly savings that increase by a fixed amount, rows of seats in a stadium, or stacks of objects with a regular pattern.
In Class 10, you learn to identify arithmetic progressions, find any term of the sequence without listing all previous terms, and calculate the sum of a given number of terms. These skills connect algebra with real-world patterns involving regular growth or decrease.
After studying this chapter, you should be able to write the general term of an AP, determine whether a given number belongs to an AP, and find sums efficiently. These ideas also prepare you for later work in sequences, series and financial mathematics.
Key ideas
- An arithmetic progression is a list of numbers where the difference between any two consecutive terms is constant; this constant is the common difference, denoted by d.
- The first term of an AP is usually denoted by a; the sequence then runs a, a + d, a + 2d, a + 3d, and so on.
- The nth term (also called the general term) of an AP is given by aₙ = a + (n − 1)d; this lets you jump directly to any term.
- If d > 0 the AP is increasing; if d < 0 it is decreasing; if d = 0 every term equals a.
- The sum of the first n terms, denoted Sₙ, can be found using formulas involving a, d and n.
- To check whether a number belongs to an AP, set aₙ equal to that number and solve for n; if n is a positive integer, the number is a term of the AP.
- Three numbers are in AP if and only if twice the middle number equals the sum of the other two.
Formulas and facts to remember
- nth term: aₙ = a + (n − 1)d Meaning: start at a and add d exactly (n − 1) times.
- Sum of first n terms (form 1): Sₙ = n/2 × [2a + (n − 1)d] Meaning: multiply the number of terms by the average of the first and last terms (expressed using a and d).
- Sum of first n terms (form 2): Sₙ = n/2 × (a + l), where l is the last term Meaning: when the last term is known, half of n times (first + last) gives the sum.
- Relation between Sₙ and aₙ: aₙ = Sₙ − Sₙ₋₁ for n ≥ 2 Meaning: the nth term is the difference between the sum up to n terms and the sum up to (n − 1) terms.
- Condition for three numbers to be in AP: If x, y, z are in AP then 2y = x + z.
Worked examples
Example 1: Finding the 20th term
A student saves ₹50 in the first week and increases her saving by ₹10 every subsequent week. Find her saving in the 20th week.
Solution First term a = 50, common difference d = 10. Using aₙ = a + (n − 1)d with n = 20: a₂₀ = 50 + (20 − 1) × 10 a₂₀ = 50 + 19 × 10 a₂₀ = 50 + 190 = 240. Her saving in the 20th week is ₹240.
Example 2: Sum of first 15 terms
Find the sum of the first 15 terms of the AP: 7, 11, 15, 19, …
Solution Here a = 7, d = 11 − 7 = 4, n = 15. Using Sₙ = n/2 × [2a + (n − 1)d]: S₁₅ = 15/2 × [2 × 7 + (15 − 1) × 4] S₁₅ = 15/2 × [14 + 56] S₁₅ = 15/2 × 70 S₁₅ = 15 × 35 = 525. The sum of the first 15 terms is 525.
Example 3: Which term equals a given value?
In the AP 5, 9, 13, 17, …, determine whether 101 is a term. If yes, find its position.
Solution a = 5, d = 4. Let aₙ = 101. 101 = 5 + (n − 1) × 4 101 − 5 = 4(n − 1) 96 = 4(n − 1) n − 1 = 24 n = 25. Since n is a positive integer, 101 is the 25th term of the AP.
Common mistakes
- Using n instead of (n − 1) in the nth-term formula → Remember you add d only (n − 1) times after the first term.
- Forgetting the factor of 1/2 in the sum formula → Write the formula first, then substitute values carefully.
- Taking d as positive when the sequence is decreasing → Compute d = (second term) − (first term); keep the sign.
- Assuming every sequence with a pattern is an AP → Check that the difference between consecutive terms is constant throughout.
- Mixing up aₙ and Sₙ in problems → Read whether the question asks for a particular term or the sum of terms.
Quick revision
- AP: a, a + d, a + 2d, … where d is the common difference.
- nth term: aₙ = a + (n − 1)d.
- Sum of n terms: Sₙ = n/2 × [2a + (n − 1)d] or n/2 × (a + l).
- To check membership, solve aₙ = given value for n; n must be a positive integer.
- d can be positive, negative or zero.