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Time, Speed & Distance
Every CAT Time, Speed & Distance formula on one page — relative speed, trains crossing poles and platforms, boats and streams, and delay problems — each with a worked example.
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Progressions (AP, GP, HP)
CAT'26 QUANT CHEATSHEET
Every progressions formula and CAT shortcut you need for CAT 2026 — on one page.
Progressions look like pure formula-memorisation until CAT hides them inside a data-interpretation set or a disguised series that needs telescoping to crack. The three progressions — arithmetic, geometric and harmonic — share a small toolkit: find the nth term, sum the series, and know how their means relate to each other. This sheet lays out every formula you need for the CAT quant section: AP and GP basics, the arithmetic, geometric and harmonic means, the AM-GM-HM inequality, special series for natural numbers and their powers, arithmetic-geometric progressions, telescoping sums, and the term-insertion properties of AP and GP, each with a worked example in real numbers. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Progressions: every formula you need
1AP: nth Term
Each term increases by the same fixed common difference.
an = a + (n−1)d
Example: a=5, d=3 → 10th term = 5+9·3 = 32.
Common Mistake: Use (n−1)d, not nd — the first term itself has zero differences added.
2Sum of an AP
Add the first n terms of an arithmetic progression.
Sn = n/2 [2a+(n−1)d] = n/2 (a+l)
Example: a=5, d=3, n=10 → 10/2[10+27] = 5·37 = 185.
3Arithmetic Mean
The single value that balances a set of numbers evenly.
AM(a,b) = (a+b)/2 | AM of n terms = Σterms ÷ n
Example: AM of 4 and 10 = (4+10)/2 = 7.
4GP: nth Term
Each term is the previous one scaled by a fixed common ratio.
an = a · rn−1
Example: a=3, r=2 → 5th term = 3·24 = 48.
5Sum of a Finite GP
Add the first n terms of a geometric progression.
Sn = a(rn−1)/(r−1), r ≠ 1
Example: a=1, r=2, n=5 → 1(32−1)/1 = 31 (i.e. 1+2+4+8+16).
6Sum of an Infinite GP
The series converges only when the ratio's size is under 1.
S∞ = a/(1−r), only valid for |r| < 1
Example: a=1, r=1/2 → 1/(1−0.5) = 2.
Common Mistake: The infinite-sum formula only applies when |r| < 1 — check this before using it.
7Geometric Mean
The multiplicative balance point of two positive numbers.
GM(a,b) = √(ab)
Example: GM of 4 and 16 = √64 = 8.
8Harmonic Progression & Mean
An HP is a sequence whose reciprocals form an AP.
HM(a,b) = 2ab/(a+b)
Example: HM of 4 and 16 = 2·4·16/20 = 128/20 = 6.4.
CAT Hack: To handle an HP directly, invert every term to convert it into an AP first.
9AM ≥ GM ≥ HM
The three means of the same numbers are always ordered this way.
AM ≥ GM ≥ HM, equality only when all terms are equal
Example: for 4 and 16: AM=10, GM=8, HM=6.4 → 10 ≥ 8 ≥ 6.4.
CAT Favourite: A recurring CAT pattern for proving inequalities or finding bounds on an expression.
10Sum of Special Series
Closed forms for the sum of the first n naturals, squares and cubes.
Σn = n(n+1)/2 | Σn2 = n(n+1)(2n+1)/6 | Σn3 = (Σn)2
Example: n=5: Σn=15, Σn2=55, Σn3 = 152 = 225.
CAT Favourite: Σn3 = (Σn)2 — the sum of cubes is exactly the square of the sum.
11Arithmetic-Geometric Progression (AGP)
A term-by-term product of an AP and a GP, summed to infinity.
S∞ = a/(1−r) + dr/(1−r)2, |r| < 1
Example: 1 + 2(½) + 3(¼) + … (a=1,d=1,r=½) → 2 + 2 = 4.
CAT Insight: Recognise an AGP whenever each term is (linear in n) × (a fixed ratio to the power n).
12Telescoping Sums
Split a term into a difference so consecutive parts cancel out.
1/(n(n+1)) = 1/n − 1/(n+1)
Example: Σ from n=1 to 99 of 1/(n(n+1)) = 1 − 1/100 = 99/100.
CAT Hack: Whenever a sum has a product like n(n+1) in the denominator, try splitting it into a telescoping difference.
13Three Terms in AP
The middle term is always the average of its neighbours.
a, b, c in AP ⇔ 2b = a + c
Example: 3, x, 11 in AP → 2x = 14 → x = 7.
14Three Terms in GP
The middle term squared equals the product of its neighbours.
a, b, c in GP ⇔ b2 = ac
Example: 3, x, 27 in GP → x2=81 → x = 9.
15Number of Terms in an AP
Work out how many terms an AP contains from its first and last term.
n = [(l−a)/d] + 1
Example: AP from 5 to 50, d=5 → (45/5)+1 = 10 terms.
CAT exam shortcuts, traps & revision
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CAT Exam Shortcuts
- an = a+(n−1)d for AP; an = a·rn−1 for GP
- Sn = n/2(a+l) is often faster than the full AP sum formula
- Invert an HP into an AP before doing any arithmetic on it
- AM ≥ GM ≥ HM always, with equality only when every term is equal
- Σn, Σn2, Σn3 are worth memorising outright
- For 3 terms in AP: 2b=a+c; in GP: b²=ac
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Most Common CAT Traps
- Using n instead of (n−1) in the AP or GP nth-term formula.
- Applying the infinite GP sum formula when |r| is not less than 1.
- Getting the AM-GM-HM order backwards.
- Forgetting to invert terms before treating a sequence as an HP.
- Confusing Σn2 with (Σn)2 — they are not the same thing (only Σn3 equals the latter).
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30-Second Revision Box
- AP: an=a+(n−1)d; Sn=n/2(a+l)
- GP: an=a·rn−1; S∞=a/(1−r) for |r|<1
- AM=(a+b)/2; GM=√(ab); HM=2ab/(a+b)
- AM ≥ GM ≥ HM, equal only when all terms match
- Σn=n(n+1)/2; Σn2=n(n+1)(2n+1)/6; Σn3=(Σn)²
- 3 in AP: 2b=a+c; 3 in GP: b²=ac
Progressions rewards recognising the pattern over memorising every variant — once a series is identified as an AP, GP, or a disguised telescoping sum, the formula is usually one substitution away. Drill this sheet until the AM-GM-HM order and the special series are reflex, then test them on full sets and track progress with the CAT score predictor. It pairs directly with the Number System cheat sheet, since the special series and summation identities overlap. For more guides, browse the Optima Learn blog or explore every study guide, work through the CAT exam hub, and when you want mentor-led prep, book a free CAT 2026 call.
