Arithmetic Progressions - Quick Notes & Formula Sheet
Class 10 CBSE Mathematics · Chapter 5 · Complete Revision Guide (2026-27)
Syllabus Note for 2026-27
This chapter's core focus is mastery of the nth term and sum of n terms formulas - CBSE consistently emphasizes these two skills above everything else here. Real-life word problems (salary increments, savings patterns, stadium seating, etc.) using both formulas remain part of the current NCERT Reprint 2026-27 scope, so don't skip practicing them even if you've seen older material suggesting otherwise.
1. What is an Arithmetic Progression?
An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant is called the common difference (d).
General AP: a, a+d, a+2d, a+3d, ...
Here a is the first term. To check if a sequence is an AP, verify that (term2 - term1) = (term3 - term2) = (term4 - term3), and so on - the difference must stay exactly the same throughout.
Trap: Forgetting d can be negative
An AP doesn't have to increase. If d is negative, the sequence decreases (e.g., 20, 15, 10, 5, ... has d=-5). Students sometimes assume "arithmetic progression" always means increasing - it doesn't.
2. nth Term of an AP (Highest Priority Formula)
a_n = a + (n-1)d
Here a_n is the value of the nth term, a is the first term, n is the term's position number, and d is the common difference.
Trap: Using n instead of (n-1)
This is the single most common calculation slip in this chapter. The formula multiplies d by (n-1), not by n. For the very first term (n=1), this correctly gives a_1 = a + 0 = a. Writing a_n = a + nd instead throws off every term by exactly one step.
Quick check tip
To find which term of an AP equals a given value, set a_n equal to that value and solve for n. If n doesn't come out as a positive whole number, that value isn't actually a term of this AP.
3. Sum of the First n Terms
S_n = n/2 × [2a + (n-1)d]
Use this version when you know the first term (a), common difference (d), and number of terms (n), but not necessarily the last term.
S_n = n/2 × (a + l)
Use this simpler version instead when the last term (l) is already known - it avoids recalculating d entirely.
Trap: Mixing up which sum formula to use
If a question already gives you the last term directly, using the longer S_n = n/2[2a+(n-1)d] formula instead of the shorter S_n = n/2(a+l) wastes time and increases the chance of arithmetic slips. Check what's given before choosing your formula.
4. Relationship Between a_n and S_n
a_n = S_n - S_(n-1)
This lets you find any individual term if you already know the sum formula (or values) for that many terms and one fewer terms - useful in HOTS-style questions that give you S_n as an expression in n.
Trap: Forgetting this only works for n≥2
For the first term specifically, a_1 = S_1 (there's no "S_0" to subtract). Only for n=2 onward does a_n = S_n - S_(n-1) apply.
5. Word Problem Patterns (High Exam Weightage)
| Problem Type | Key Setup |
|---|---|
| Salary/savings that increases by a fixed amount each year | a = starting value, d = fixed yearly increase; use a_n for a specific year, S_n for total over several years |
| Stadium/auditorium seating (rows increasing by a fixed number) | a = seats in row 1, d = extra seats per row; S_n gives total seats across n rows |
| Distance/rows of trees, poles, or objects placed at regular intervals | Often requires doubling a distance-and-return trip; read carefully whether the question wants one-way or round-trip total |
| Debt repayment in equal or increasing installments | Track remaining balance as first term, reducing installment as d (often negative) |
Trap: Not identifying a and d correctly from the story
Word problems often describe the pattern in a roundabout way (e.g., "Reena's salary starts at Rs.8000 and grows by Rs.500 every year"). Always explicitly write out a= and d= before applying any formula - jumping straight to substitution is where most word-problem errors happen.
6. Full Recap - Every Trap in One Place
| Trap | Correct Understanding |
|---|---|
| Assuming an AP always increases | d can be negative - APs can decrease too |
| Using a_n = a+nd | Correct formula is a_n = a+(n-1)d |
| Using the longer sum formula when the last term is already known | Use S_n = n/2(a+l) instead - faster and less error-prone |
| Applying a_n=S_n-S_(n-1) for n=1 | For the first term, a_1=S_1 directly - the subtraction only applies from n=2 onward |
| Substituting into formulas before identifying a and d from the word problem | Always write out a= and d= explicitly first |
| Assuming a value must belong to a given AP without checking | Solve for n; if it isn't a positive whole number, that value isn't a term of the AP |
Frequently Asked Questions
What is the formula for the nth term of an arithmetic progression?
a_n = a + (n-1)d, where a is the first term, d is the common difference, and n is the term's position.
What is the formula for the sum of the first n terms of an AP?
S_n = n/2 × [2a+(n-1)d], or the shorter S_n = n/2 × (a+l) if the last term l is already known.
Are real-life word problems on sum of n terms included in the CBSE 2026-27 syllabus?
Yes - current NCERT Reprint 2026-27 material and CBSE exam patterns do test real-life word problems on both the nth term and sum of n terms for this chapter.
What is the most common mistake students make with the nth term formula?
Using n instead of (n-1) in a_n=a+(n-1)d, which shifts every calculated term by one position. Always double check this specific step.
Practice What You've Just Revised
This chapter follows the same expanded 5-part quiz structure (100 questions total) used for Quadratic Equations.
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