Arithmetic Progressions - Quick Notes & Formula Sheet
Class 10 CBSE Mathematics · Chapter 5 · Complete Revision Guide (2026-27)
1. What is an Arithmetic Progression (AP)?
A list of numbers in which each term (after the first) is obtained by adding a fixed number, called the common difference (d), to the previous term.
d = a2 - a1 = a3 - a2 = ... (constant throughout)
Trap: d can be negative or a fraction
Students often assume an AP must increase. A decreasing sequence (d negative) or one with fractional/decimal common difference is still a valid AP - always compute d by subtraction, don't just "look" at the sequence.
2. nth Term Formula (Highest Weightage)
an = a + (n-1)d
Where a is the first term, d is the common difference, and n is the position of the term you want.
Trap: Off-by-one error in (n-1)
The most common mistake in this entire chapter: writing an=a+nd instead of a+(n-1)d. Always double-check with n=1: a1 should equal a exactly, which only works with the (n-1) form.
3. Sum of First n Terms
Sn = n/2 × [2a + (n-1)d]
Sn = n/2 × (a + l), where l is the last term
Use the second form when the last term is already known or easy to find - it's faster than recalculating (n-1)d.
Trap: Confusing an and Sn
A very common slip: using the sum formula when the question asks for a specific term, or vice versa. Read the question carefully - "find the 15th term" needs an; "find the sum of the first 15 terms" needs Sn.
4. Finding a Term from the End
To find the nth term from the end of an AP with last term l: treat l as the "first term" of a reversed AP with common difference -d.
nth term from the end = l - (n-1)d
5. Word Problem Patterns (Very High Exam Weightage - Still Fully Examinable in 2026-27)
| Problem Type | Key Setup |
|---|---|
| Salary/savings increasing by fixed amount yearly | a = starting value, d = fixed yearly increase |
| Rows of seats/bricks/logs (decreasing pattern) | a = first row's count, d = negative (decreasing) |
| Total value after n years/terms | Use Sn, not an - question asks for a total, not a single year's value |
| "In which year does X happen" | Set an equal to the target value, solve for n |
Quick check tip
Before solving, underline the exact phrase in the question - "value in the nth year" needs an; "total value after n years" needs Sn. This single habit prevents the most common word-problem mistake in this chapter.
6. Full Recap - Every Trap in One Place
| Trap | Correct Understanding |
|---|---|
| Assuming AP must be increasing | d can be negative, zero, or a fraction |
| Writing an=a+nd | Correct formula is an=a+(n-1)d |
| Using Sn when question asks for a single term | Read the exact wording - "value in year n" vs "total after n years" |
| Forgetting n must be a positive integer when solving for it | If solving an=a+(n-1)d for n gives a non-integer or negative value, re-check the problem setup |
Frequently Asked Questions
What is the formula for the nth term of an AP?
an = a + (n-1)d, where a is the first term, d is the common difference, and n is the term number.
What is the formula for the sum of the first n terms?
Sn = n/2 × [2a+(n-1)d], or equivalently Sn = n/2 × (a+l) using the last term l.
Are word problems on APs still tested in CBSE 2026-27?
Yes - real-life application word problems remain fully examinable this year, alongside the core nth term and sum formulas.
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