Chapter 5: Arithmetic Progressions
1. What is an Arithmetic Progression (AP)?
An Arithmetic Progression is a list of numbers in which each term (except the first) is obtained by adding a fixed number to the preceding term. This fixed number is called the common difference (d). It can be positive, negative or zero.
Common difference: d = ak+1 − ak (same for every k)
- a = first term, d = common difference, an = nth term.
- A finite AP has a last term (l); an infinite AP does not.
- To check whether a list is an AP: compute a2 − a1, a3 − a2, a4 − a3 ... — all must be equal.
Examples: 2, 5, 8, 11, ... (d = 3); 10, 7, 4, 1, ... (d = −3); 5, 5, 5, ... (d = 0). But 1, 2, 4, 8, ... is NOT an AP (differences 1, 2, 4 are unequal).
2. nth Term (General Term) of an AP
Here n is a positive integer. an is also called the general term. If there are m terms, then am = l, the last term.
nth term from the end (of a finite AP with last term l): l − (n − 1)d. Equivalently, reverse the AP (first term l, common difference −d) and use the usual formula.
Worked example: Which term of the AP 21, 18, 15, ... is −81?
a = 21, d = −3. an = −81 ⇒ 21 + (n − 1)(−3) = −81 ⇒ (n − 1)(−3) = −102 ⇒ n − 1 = 34 ⇒ n = 35. So −81 is the 35th term.
3. Sum of First n Terms of an AP
Sn = n/2 (a + l), when the last term l is known
Sum of first n positive integers: 1 + 2 + ... + n = n(n + 1)/2
Very important relation: the nth term equals the difference of consecutive sums:
Worked example: Find the sum of the first 22 terms of the AP 8, 3, −2, ...
a = 8, d = −5, n = 22.
S22 = 22/2 [2(8) + 21(−5)] = 11[16 − 105] = 11 × (−89) = −979.
Worked example (given Sn formula): If Sn = 4n − n2, then S1 = 3 = a1; S2 = 4 ⇒ a2 = S2 − S1 = 1; so d = 1 − 3 = −2, and an = Sn − Sn−1 = 5 − 2n.
4. Selecting Terms of an AP (Useful Trick)
| Number of terms | Terms to assume | Common difference |
|---|---|---|
| 3 | a − d, a, a + d | d |
| 4 | a − 3d, a − d, a + d, a + 3d | 2d |
| 5 | a − 2d, a − d, a, a + d, a + 2d | d |
Use these when the sum of terms is given — the d terms cancel on adding. E.g., "The sum of three numbers in AP is 24 and their product is 440": a − d + a + a + d = 24 ⇒ a = 8; then 8(8 − d)(8 + d) = 440 ⇒ 64 − d2 = 55 ⇒ d = ±3 ⇒ numbers are 5, 8, 11.
5. Other Key Facts
- Three numbers a, b, c are in AP if and only if 2b = a + c (b is the arithmetic mean of a and c).
- If a fixed number is added to / subtracted from every term of an AP, it remains an AP with the same d. If every term is multiplied by k, it remains an AP with common difference kd.
- Difference between mth and nth terms: am − an = (m − n)d.
- If m times the mth term equals n times the nth term, then the (m + n)th term is 0 (classic 3-mark result).
6. Applications in Daily Life (Retained in Syllabus)
Board papers regularly frame AP word problems from real situations:
- Salary/savings: starting salary a with fixed annual increment d — salary in nth year = a + (n − 1)d; total earned in n years = Sn.
- Instalments: loan repaid in instalments increasing/decreasing by a fixed amount.
- Production: factory output increasing uniformly — "production in 3rd year is X, in 7th year is Y; find first-year production, annual increase, and total production in 7 years."
- Ladder rungs, flower beds, seats in rows, logs stacked in rows (each row has a fixed number more/less than the previous), penalty for delay increasing daily, prize money decreasing uniformly.
7. Common Board Question Patterns
- Find an, d, or the number of terms; which term is a given number; nth term from the end (1–2 marks).
- Find the middle term(s) of a finite AP; how many two/three-digit numbers are divisible by a given number (2–3 marks).
- Find Sn; find n when Sn is given; problems using an = Sn − Sn−1 (3 marks).
- Two-condition problems: "4th term is ..., 8th term is ..., find the AP" — form two equations in a and d and solve (3 marks).
- Daily-life case studies: stacking, instalments, salary increments, auditorium seats (4–5 marks).