Inequalities Advanced: AM-GM, Cauchy-Schwarz and Optimization Problems

A ball drops from 10 metres. Each bounce carries it back to 60 percent of the height it just fell from, and it keeps bouncing until it stops. Find the total distance it travels. Most aspirants picture a curve and reach for something geometric. There is no curve here. The heights form a geometric progression, and the whole thing is one infinite GP sum that lands on 40 metres in about fifteen seconds. That is exactly why AP GP disguised CAT questions sink strong students. You know every progression formula cold, yet the word problem never tells you a progression is hiding inside it.

This guide closes that gap. We cover eight ways CAT hides AP, GP, and HP inside ordinary stories, give the one recognition signal for each, then hand you a problem set where your only job is to name the progression before you solve.

Why you miss the progression

Standard prep trains one half of the skill. The questions already say "the terms of an AP" or "in a GP", so you go straight to summing or finding the nth term. You get fast at the arithmetic and never build the muscle that matters more: spotting that a salary, a bounce, or a stack of logs is a progression at all.

CAT writers know this. They strip the label out and bury the structure inside a scene: a machine loses value, a pendulum slows, money compounds. The math is identical to the labelled version, but the trigger word is gone. The difference between strong and average Quant here is rarely the formula. It is the reading. Aspirants who treat arithmetic over algebra in CAT Quant as a first instinct catch these patterns faster, because they scan for clean numeric relationships instead of setting up equations.

The fix is a habit, not a new formula. Before you compute, ask one thing: how does each value reach the next? Everything follows from the answer.

The one signal that names the progression

Three relationships cover almost every progression CAT throws at you, and each has a single tell.

• Fixed difference means AP. Each step adds or subtracts the same amount: a salary rising by 2000 a year, a row of logs with one fewer than the row below, an installment growing by a set figure each month.
• Fixed ratio means GP. Each step multiplies by the same factor: a ball rebounding to 60 percent of its height, money at 10 percent compound, a machine depreciating 20 percent a year.
• Equal distance at different speeds means HP. When you average rates over equal stretches, the speeds form a harmonic progression and the average is their harmonic mean. The reciprocals of an HP are an AP, the cleanest way to handle it.

Addition points to AP, multiplication to GP, equal-distance averaging to HP. Ask which is happening and the formula picks itself, the same recognition-first discipline that powers clean work in advanced inequalities for CAT.

The 8 disguises CAT uses

These are the eight costumes a progression most often wears, each with its scene and its signal.

1. Bouncing ball heights (GP)

A ball rebounds to a fixed fraction of its previous height, so the heights form a GP. Total distance is the first drop plus twice the rebound sum. A 10 metre drop, ratio 0.6, gives 10 + 2(10)(0.6)/(1 - 0.6) = 40 metres. Signal: "fraction of the previous height".

2. Equal-distance average speed (HP)

You cover the same distance at two different speeds and want the average. The speeds form a harmonic progression, so the answer is their harmonic mean. One way at 60 and back at 40 gives 2(60)(40)/(60 + 40) = 48, not 50. Signal: "same distance, different speeds".

3. Compound interest growth (GP)

Compounding money is multiplied by the same factor each period, a GP with ratio (1 + r). A deposit of 10000 at 10 percent for three years becomes 10000 times 1.1 cubed = 13310. Signal: percent growth applied repeatedly.

4. Salary or installment increments (AP)

Pay rising by a fixed amount each year, or an installment adding a set sum each month, is a textbook AP. A salary of 30000 with a 2000 yearly raise earns 170000 over five years, the AP sum. Signal: "increases by a fixed amount".

5. Depreciation (GP)

An asset losing the same percent of its value each year shrinks by a fixed ratio, a GP with ratio (1 - r). A machine worth 50000 depreciating at 20 percent is worth 50000 times 0.8 cubed = 25600 after three years. Signal: "loses a percentage of value", the mirror of compound growth.

6. Stacked rows and logs (AP sum)

Logs or pipes stacked so each row holds one fewer than the row below form an AP. A pile with 20 in the bottom row, dropping by one across 12 rows, holds 6 times (40 - 11) = 174 logs. Signal: a stack with a constant step between rows.

7. Pendulum or swing arcs (GP)

A pendulum loses energy each swing, so each arc is a fixed fraction of the last, a GP. A first arc of 24 cm, each later arc 5/6 of the previous, totals 24/(1 - 5/6) = 144 cm. Signal: "each swing a fraction of the last", the same shrinking factor as the ball.

8. Insect and repeated-fraction distance (infinite GP)

The repeated-fraction journey is a true infinite GP. An insect flying a first leg of 12 metres, each later leg 3/4 of the previous, covers 12/(1 - 3/4) = 48 metres. Signal: a process repeating forever with a fixed shrinking step.

Recognition-signal table

Keep this table in your head. In the exam, the signal should fire on its own the moment you read the scene.

Identify-then-solve: 12 problems

Cover the answers. Name the progression first, then solve. The naming is the skill; the arithmetic is easy once the label is right.

Name the progression correctly on ten or more before solving and your recognition is exam-ready. If the labels gave you trouble, the gap is reading, not formulas, the cheaper one to fix. A fuller bank of progression practice questions with timed recognition closes it fastest.

Where recognition goes wrong

Many progression problems hide a number-theory layer, where the step or term count turns on factors and multiples. Pairing this skill with sharp HCF and LCM tricks for CAT handles divisibility-flavoured progressions, like the multiples-of-7 sum, with ease.

Common questions on disguised progressions

Progressions reward the reader, not the memoriser. Once the bounce, the salary, and the depreciating machine read as the same three patterns underneath, this whole family stops being a guessing game. Build the habit on every Quant set, then track how a sharper section moves your percentile with the CAT score predictor before your next mock. For a structured run through every Quant topic, the full set of CAT preparation guides takes it from here.