Whether it is the seating rows in a stadium or the bouncing of a ball, numbers that follow a clear pattern are everywhere. In the NDA exam, Sequences and Series is one of the most scoring chapters because the questions are formula-driven and predictable. Learn a handful of formulas well, and you can lock down 3–5 easy marks every single year.
Why This Chapter Is a Scoring Goldmine
Every NDA Maths paper carries questions from Sequences and Series, and they are usually direct, formula-based, and quick to solve. Unlike calculus or trigonometry, you rarely need long manipulation — once you spot the pattern, the answer drops out in two or three lines. For a student fighting against the clock in a 150-question paper, that speed is gold.
The topic also overlaps with other chapters: binomial sums, mean problems in statistics, and even compound-interest style growth all rest on the same ideas. So the effort you put in here quietly boosts your performance in several other areas too. That is why Cavalier teachers insist on building a rock-solid foundation in progressions before moving on to the heavier chapters.
The good news is that the entire chapter rests on just three families of progressions and roughly a dozen formulas. Once these are at your fingertips, most questions become a matter of plugging in numbers carefully. The aim of this lesson is to make every one of those formulas feel natural rather than memorised by rote.
First identify the type of progression (AP, GP or HP). Half the battle is choosing the correct formula; the arithmetic is the easy part.
Sequence, Series and Progression
A sequence is an ordered list of numbers following a rule, written as a1, a2, a3, …. Each number is called a term.
A series is what you get when you add the terms of a sequence: a1 + a2 + a3 + …
A progression is a special sequence whose terms follow a definite mathematical rule, such as a constant difference (AP) or a constant ratio (GP).
Sequence = the list. Series = the sum of the list. A finite sequence has a last term; an infinite sequence does not.
Arithmetic Progression (AP)
An AP is a sequence where each term differs from the previous one by a fixed number called the common difference (d). If the first term is a, the terms are: a, a + d, a + 2d, a + 3d, …
nth term: an = a + (n − 1)d
Sum of n terms: Sn = (n/2)[2a + (n − 1)d]
Also: Sn = (n/2)(a + l), where l is the last term.
A handy fact: three numbers are in AP if and only if the middle term equals the average of the other two, i.e. 2b = a + c. This single idea answers a surprising number of objective questions, because examiners love giving you three terms and asking whether they form an AP.
Think of an AP as climbing a staircase where every step has the same height. The first term tells you where you start, and the common difference tells you the size of each step. To reach the nth step you take (n − 1) steps of height d, which is exactly why the term formula has (n − 1)d and not nd. Keeping this picture in mind prevents the most common slip in the chapter.
To assume three numbers in AP, take them as a − d, a, a + d. For four numbers, use a − 3d, a − d, a + d, a + 3d. This trick makes the sum simplify instantly.
Geometric Progression (GP)
A GP is a sequence where each term is obtained by multiplying the previous one by a fixed number called the common ratio (r). The terms are: a, ar, ar2, ar3, …
nth term: an = ar(n−1)
Sum of n terms (r ≠ 1): Sn = a(rn − 1)/(r − 1)
Sum of infinite GP (when |r| < 1): S∞ = a/(1 − r)
Three numbers are in GP if and only if the square of the middle term equals the product of the other two, i.e. b2 = ac. This is why the middle term is called the geometric mean. Where an AP grows by repeated addition, a GP grows or shrinks by repeated multiplication, which is exactly how populations, bacteria and money under compound interest behave.
When the common ratio lies between 0 and 1, the terms get smaller and smaller and the GP tends towards zero. This is the situation that makes an infinite sum meaningful: each new term adds less than the one before, so the total settles at a fixed value. A bouncing ball that rises to a fraction of its previous height after every bounce is a classic real-world infinite GP.
The infinite-sum formula works only when |r| < 1. If |r| ≥ 1 the terms keep growing (or stay constant), so the sum does not settle to a finite value.
Harmonic Progression (HP)
A sequence is in Harmonic Progression if the reciprocals of its terms form an AP. For example, 1, 1/2, 1/3, 1/4, … is an HP because 1, 2, 3, 4, … (the reciprocals) form an AP.
To find the nth term of an HP, write the reciprocals as an AP, find that AP's nth term, then take its reciprocal.
The harmonic mean (HM) of two numbers a and b is HM = 2ab/(a + b). There is no simple direct formula for the sum of an HP, so NDA questions on HP usually ask for a particular term, not a sum.
The harmonic mean shows up naturally whenever you average rates or speeds. For instance, if you travel a fixed distance at one speed and return at another, your average speed for the whole trip is the harmonic mean of the two speeds, not the ordinary arithmetic average. Spotting that an HP question is really a disguised AP question is the key skill the exam tests here.
Arithmetic, Geometric and Harmonic Means
For two positive numbers a and b:
- Arithmetic Mean (AM) = (a + b)/2
- Geometric Mean (GM) = √(ab)
- Harmonic Mean (HM) = 2ab/(a + b)
For any two positive numbers: AM ≥ GM ≥ HM. Equality holds only when a = b.
Also a beautiful relation: GM2 = AM × HM, so GM is the geometric mean of AM and HM too.
"Inserting n arithmetic means" between a and b means building an AP with a and b as the end terms and n terms squeezed in between, giving (n + 2) terms in all. The same idea works for geometric means with a GP.
The AM ≥ GM ≥ HM chain is a favourite NDA trick. If a question asks which mean is largest for unequal numbers, the answer is always AM.
Sums of Special Series
Some sums appear again and again in the NDA exam, so memorise them cold. For the first n natural numbers:
Sum of first n natural numbers: Σn = n(n + 1)/2
Sum of squares: Σn2 = n(n + 1)(2n + 1)/6
Sum of cubes: Σn3 = [n(n + 1)/2]2
Notice the neat result that the sum of the first n cubes equals the square of the sum of the first n natural numbers. That is, Σn3 = (Σn)2.
The sum of the first n odd numbers is simply n2, and the sum of the first n even numbers is n(n + 1). These shortcuts save precious seconds.
You will often meet a series that is none of these standard forms but can be split into them. The trick is to write the general term, then break the whole sum into separate Σn2, Σn and Σ1 pieces, each of which you already know. This method of splitting by the general term turns a frightening-looking series into three familiar sums added together.
When a series mixes powers, always write the nth term first. Once you have the general term, the sigma formulas do the rest of the work for you.
Worked Example: Mixing AP and GP Ideas
Let us solve a typical multi-step problem the way you should in the exam hall.
The sum of the first three terms of a GP is 39 and their product is 729. Find the three terms.
Choosing the terms as a/r, a, ar made the product collapse to a single unknown — that is the move examiners reward.
Common Mistakes to Avoid
Most lost marks in this chapter come from small slips, not hard concepts.
- Using the AP sum formula on a GP (or vice versa) — always classify the progression first.
- Forgetting the (n − 1) in the nth-term formulas and writing nd instead of (n − 1)d.
- Applying S∞ = a/(1 − r) when |r| ≥ 1.
- For HP, trying to add terms directly instead of switching to the reciprocal AP.
When you take three numbers in AP as a − d, a, a + d, do not forget that d can be negative. Both answers may be valid — check against the question's conditions.
Previous-Year Style Practice
Here is a question modelled on the NDA exam pattern. Try it before reading the solution.
Q. If the 5th term of an AP is 11 and the 9th term is 7, what is the sum of its first 15 terms?
Answer: From a + 4d = 11 and a + 8d = 7, subtracting gives 4d = −4, so d = −1 and a = 15. Then S15 = (15/2)[2(15) + 14(−1)] = (15/2)(30 − 14) = (15/2)(16) = 120.
Notice how subtracting the two term-equations instantly removed a and gave d. This elimination trick is the fastest route in nearly every "two terms given" AP problem.
Quick Revision Before the Exam
Glance over these the night before your paper and the morning of it.
- AP: an = a + (n − 1)d; Sn = (n/2)[2a + (n − 1)d].
- GP: an = ar(n−1); Sn = a(rn − 1)/(r − 1); S∞ = a/(1 − r) only if |r| < 1.
- HP: take reciprocals → AP → solve → reciprocal back.
- AM ≥ GM ≥ HM, with GM2 = AM × HM.
- Σn = n(n+1)/2, Σn2 = n(n+1)(2n+1)/6, Σn3 = (Σn)2.
Practise 10–15 mixed problems daily for a week before the exam. Speed in this chapter directly frees up time for the harder calculus and geometry sections.
Frequently asked questions
What is the difference between a sequence and a series?
A sequence is an ordered list of numbers that follow a rule, such as 2, 4, 6, 8. A series is the sum of those terms, such as 2 + 4 + 6 + 8. In short, the series is what you get when you add up a sequence.
How many questions come from Sequences and Series in the NDA exam?
Typically 3 to 5 questions appear in each NDA Maths paper, and they are usually direct formula-based problems. This makes it one of the most reliable scoring chapters if you have memorised the standard formulas.
When can I use the infinite GP sum formula a/(1 - r)?
Only when the common ratio satisfies |r| < 1, that is, r lies strictly between -1 and 1. If |r| is 1 or larger, the terms do not shrink, so the infinite sum does not converge to a finite value.
Which mean is the largest: AM, GM or HM?
For any two distinct positive numbers, the arithmetic mean is the largest and the harmonic mean is the smallest, following AM > GM > HM. They become equal only when the two numbers are identical.
What is the quickest way to handle a Harmonic Progression?
Convert the HP into an AP by taking the reciprocal of each term. Solve the problem in AP form using standard formulas, then take the reciprocal of your result to return to the HP value.
Related NDA Maths topics
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