CAT Quant · Study & Practice

Arithmetic Progression

AreaAlgebra DifficultyEasy–Moderate CAT weightage1–3 questions (directly + inside Series, Averages, Logical-arrangement and DI sets)

An arithmetic progression (AP) is a sequence in which each term differs from the one before it by a fixed amount called the common difference. That single idea — constant difference — is one of the most quietly useful tools in CAT Quant. Whenever a question hides a pattern that grows by equal steps (seats per row in an auditorium, instalments that rise by a fixed sum, days needed when output climbs steadily, even the count of multiples of a number in a range), an AP is sitting underneath it. CAT rarely asks you to "find the 10th term" in plain words; it dresses the AP up as a word problem and rewards students who can spot the linear pattern, pick the right formula, and avoid arithmetic slips. This chapter builds that fluency: the nth-term formula a_n = a + (n−1)d, the two faces of the sum formula S_n = n/2[2a + (n−1)d] = n/2(first + last), the average-of-neighbours property, and the symmetric-term trick (a−d, a, a+d) that collapses messy simultaneous equations into one line. Each topic comes with worked CAT-style examples, the fastest method, and the traps that cost careless aspirants easy marks.

Topics

1 nth Term 4 worked · 5 practice · 8-Q test Start → 2 Sum of n Terms 4 worked · 5 practice · 8-Q test Start → 3 AP Applications 4 worked · 5 practice · 8-Q test Start →

⚡ CAT shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • Read d straight off the sequence (any term minus the previous one) — never count up term by term.
  • For evenly spaced lists, count terms with n = (last − first)/d + 1; this nails "how many multiples" questions.
  • Two terms given? Subtract their equations to kill a and get d instantly: a_p − a_q = (p − q)d.
  • Sum fast with S_n = n/2 (first + last) = n × (average term) whenever both end terms are known.
  • Given the sum (and product) of terms in AP, centre them: 3 terms (a − d, a, a + d), 4 terms (a − 3d, a − d, a + d, a + 3d).
  • Keep 1 + 2 + … + n = n(n + 1)/2 and 1 + 3 + … + (2n − 1) = n² on instant recall for series sums.

⚠️ Common mistakes & traps

CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Using a + nd for the nth term instead of a + (n − 1)d (off-by-one on the count of d added).
  • Forgetting the "+1" in n = (last − first)/d + 1, which undercounts the terms by one.
  • Writing unknown terms as a, a + d, a + 2d for sum-product problems instead of the symmetric form, which forces messy quadratics.
  • Taking the common difference of four symmetric terms as d instead of 2d (the spacing in a − 3d, a − d, a + d, a + 3d is 2d).
  • Dropping the negative root of n in a sum equation, or keeping a non-integer n — the number of terms must be a positive integer.

📈 CAT exam insight & PYQ analysis

AP appears in CAT and the OMET exams (XAT, SNAP, IIFT) less as a stand-alone series question and more as the hidden skeleton of word problems — instalments that rise by a fixed amount, seats or logs stacked in a pattern, counting evenly spaced multiples, or three/four unknowns whose sum and product are given. The recurring high-value patterns are the symmetric-term setup for sum-product systems, the n = (last − first)/d + 1 count for multiples, and recovering a term from S_n via a_n = S_n − S_(n−1). Difficulty stays Easy–Moderate, so these are marks you should bank quickly and accurately rather than agonise over.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

nth term of an AP?Tap to reveal
a_n = a + (n − 1)d
Sum of n terms (full form)?Tap to reveal
S_n = n/2 [2a + (n − 1)d]
Sum of n terms using end values?Tap to reveal
S_n = n/2 (first + last)
Number of terms in an evenly spaced list?Tap to reveal
n = (last − first)/d + 1
How is each term related to its neighbours?Tap to reveal
a_n = (a_(n−1) + a_(n+1))/2 — the average of them
Best way to write 3 terms in AP (sum given)?Tap to reveal
(a − d), a, (a + d)
Best way to write 4 terms in AP?Tap to reveal
(a − 3d), (a − d), (a + d), (a + 3d), common difference 2d
Recover a_n from the sum S_n?Tap to reveal
a_n = S_n − S_(n−1)
Sum 1 + 2 + … + n?Tap to reveal
n(n + 1)/2
Sum of first n odd numbers?Tap to reveal
n²
mth term from the end of an AP with last term l?Tap to reveal
l − (m − 1)d
d from two terms a_p and a_q?Tap to reveal
d = (a_p − a_q)/(p − q)

📌 Quick revision

An AP is fixed by a (first term) and d (common difference). The nth term is a + (n − 1)d; never forget the "− 1". Count terms with n = (last − first)/d + 1. Sum with S_n = n/2 [2a + (n − 1)d] = n/2 (first + last) = n × average. Each term is the average of its neighbours, so the middle term of three in AP is their mean. For sum-product systems, centre the unknowns — (a − d, a, a + d) or (a − 3d, a − d, a + d, a + 3d) — so the d-terms cancel and a falls out at once. Recover a single term from the sum via a_n = S_n − S_(n−1). Keep 1 + 2 + … + n = n(n + 1)/2 and the odd-number sum n² on instant recall.

Chapter test

📝 Arithmetic Progression — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
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🏆 Vidaara CAT success checklist

You have truly mastered Arithmetic Progression when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (3 topics)3/3
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall12 cards