AP, GP and Series in CAT Quant: 4 Patterns That Repeat

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Add up the fractions 1 over 1 times 2, plus 1 over 2 times 3, plus 1 over 3 times 4, all the way to 1 over 99 times 100. Doing it term by term would take most of the section. Doing it with the right pattern takes about fifteen seconds, and the answer is 99 over 100.

That gap between ten minutes and fifteen seconds is what separates aspirants in the progressions chapter. The formulas are short. The skill is recognising which kind of series you are looking at. Four patterns keep coming back in AP, GP and series questions, and this guide shows each with its cue, its shortcut and a worked case.

Why Series Questions Look Harder Than They Are

Progressions are a compact part of algebra. In past papers the chapter has appeared as a small number of questions alongside logarithms, equations, inequalities and functions, though CAT does not publish a chapter wise split and the count varies. What stays constant is the style: a short setup with an unusual twist.

The twist is where aspirants lose time. The question might hide the AP inside a word problem about instalments, or present a series whose terms look random until you take differences. The formulas for an AP or GP take two minutes to learn. Recognising the series under a layer of wording is the part that needs practice.

The fix is a short test you run on any series. Look at the difference between consecutive terms, then the ratio, then ask whether the terms shrink in a way that suggests cancelling. These three looks take ten seconds and point to the pattern.

It also helps to know that each pattern has a family of disguises. An AP can appear as steps on a staircase, as a salary that rises by a fixed amount each year, or as a quadratic in disguise. A GP can appear as repeated percentage growth or as a ball bouncing back to a fraction of its height. The wording changes while the structure stays the same.

The Four Pattern Test: Name the Series First

Use this as a mental checklist. Each line has a cue and a move, so by the end of the test you know what you are holding.

The order of the test is deliberate. Differences and ratios are the cheapest checks, so they come first. Telescoping needs you to look at the structure of each term, and power sums need you to recognise a familiar list. If the first two checks fail on the opening three terms, spend ten seconds on the structure before reaching for anything else.

The first two are definitions, and the last two are recognitions. Telescoping and power sums are the patterns that look unfamiliar the first time and routine the third time. That is why the later sections give them more room.

Equal Spacing: AP and the Average Trick

An arithmetic progression has a first term a and a common difference d. The nth term is a plus (n minus 1) d, and the sum of n terms is n over 2 times (first plus last). The second form is the one to keep. It says the sum is the count times the average, and it avoids carrying d through the algebra.

Counting terms is a frequent slip. For 7, 13, 19 and so on up to 187, the number of terms is (187 minus 7) divided by 6, plus 1, which is 31. Forgetting the plus 1 is the classic off by one error, so test it on a tiny case such as 7, 13, 19 to be sure.

Multiples work the same way. The sum of the first 50 multiples of 3 is 3 plus 6 and so on up to 150, so the sum is 50 over 2 times (3 plus 150), which is 3,825.

If a question gives you the sum as a quadratic, such as 3n squared plus 2n, the first term is the sum at n equals 1, which is 5. The second term is S of 2 minus S of 1, which is 16 minus 5, which is 11, so d is 6.

A neat property is worth remembering. If three numbers are in AP, the middle one is their average. If five numbers are in AP, the middle one is their average too. That makes problems with a given sum and a symmetric arrangement almost trivial, because you let the middle term be a and the others be a minus d and a plus d.

Equal Ratio: GP, Infinite Sums and Recurring Decimals

A geometric progression has first term a and common ratio r. The nth term is a times r to the power (n minus 1), and the finite sum is a times (r to the power n minus 1) over (r minus 1). For 2, 6, 18, 54, 162 the ratio is 3 and the sum is 2 times (243 minus 1) over 2, which is 242.

The infinite sum exists only when the ratio lies strictly between minus 1 and 1. The sum then equals a over (1 minus r). For 8 plus 4 plus 2 and so on, the ratio is one half and the sum is 8 over one half, which is 16. A question that gives a ratio of 2 and asks for an infinite sum has no answer, so check the ratio first.

Recurring decimals are a GP in disguise. The decimal 0.727272 and so on is 72 over 100, plus 72 over 10,000, and so on. The first term is 0.72 and the ratio is 0.01, so the sum is 0.72 over 0.99, which is 72 over 99, or 8 over 11. The pattern is that a repeating block over as many nines as the block has digits equals the decimal.

If three numbers are in GP, the square of the middle one equals the product of the other two. Use that when the question gives three numbers with a missing term. For positive numbers the arithmetic mean is at least the geometric mean, which appears in optimisation questions under the algebra chapter and is worth a quick revision.

Telescoping and Power Sums: The Patterns That Collapse

Telescoping series use partial fractions. The term 1 over n times (n plus 1) equals 1 over n minus 1 over (n plus 1). Write out a few terms and the middle pieces cancel in pairs. For the sum up to 99 times 100, only the first piece, 1, and the last piece, 1 over 100, survive, so the sum is 99 over 100. Variants such as 1 over n times (n plus 2) leave two pieces at each end.

Power sums are formulas to memorise. The sum of the first n natural numbers is n times (n plus 1) over 2. The sum of squares is n times (n plus 1) times (2n plus 1) over 6, and the sum of cubes equals the square of the sum of the first n numbers. For n equal to 10 these give 55, 385 and 3,025, which are worth knowing by heart as sanity checks.

The sum of the first n odd numbers is n squared. So 1 plus 3 plus 5 and so on up to 99 has 50 terms and equals 2,500. Many questions ask for the sum of even numbers, odd numbers or multiples inside a range, and splitting the range correctly is the entire challenge.

When a series does not match any of the four patterns, take successive differences. A constant first difference gives an AP. A constant second difference gives a quadratic general term, and you can fit it with three values. A difference that is itself a GP signals a combination, which is rarer but does appear.

Pattern	Cue	Key result
AP	Constant difference	Sum equals count times average of ends
GP	Constant ratio	Finite sum, or a over (1 minus r) if the ratio is a fraction
Telescoping	1 over n times (n plus 1) type terms	Only first and last pieces remain
Power sums	n, n squared, n cubed, odd numbers	Standard sum formulas

A Practice Routine for Progressions

This chapter rewards short, frequent practice. Solve three or four series questions a day for two weeks and write the pattern name next to each before you solve. The habit of naming first is the training, not the solving.

Keep a small notebook with one example of each pattern and one disguised version. Review it twice a week. When a new question stumps you, add it only if it introduces a new disguise, not just a new set of numbers. A notebook of twenty good disguises beats a hundred worked questions with no structure.

Series questions can eat time if you chase the wrong pattern. Our note on avoiding silly calculation mistakes in CAT Quant pairs well with the off by one check above. When to stop is covered in time consuming versus difficult CAT Quant questions, and the series chapter has plenty of the first kind.

Options in series questions often include the sum with one term missing or double counted. Our guide on why CAT Quant options include obvious trap values explains the habit of testing the answer against a small case. To see how series fit into your attempt order, read the order to attempt CAT Quant questions.

Past papers are the best final test. Our topic wise CAT previous year questions let you check how setters have disguised each pattern.