NET's sequences and series questions cover AP and GP terms and sums, harmonic progressions, the three means, infinite geometric series and recurring decimals. You need about ten formulas. The skill is spotting which progression a question hides.
Key facts
- Mathematics is 50% of NET-Engineering and NET-Natural Sciences, and 30% of NET-Architecture
- Sequences and Series is unit 6 of the Class 11 PECTAA mathematics book (National Curriculum 2023)
- Our chapter map estimates about 6 MCQs from this chapter. NUST does not publish chapter counts
Source: NUST subject weightings, verified 22 August 2026; SailToNUST chapter map
What the chapter covers
The chapter covers arithmetic, geometric and harmonic sequences, the arithmetic, geometric and harmonic means, and the sums of finite and infinite series. It ends with converting recurring decimals into fractions, which is an infinite geometric series in disguise.
NET questions here are often worded as stories: a clock that strikes, a ball that bounces, a salary that rises each year. Strip the story and you have a, d or r, and n.
Formulas
Arithmetic progression (AP), first term a, common difference d:
a_n = a + (n - 1) d
S_n = (n/2) [2a + (n - 1) d] = (n/2)(a + l) l = last term
Geometric progression (GP), first term a, common ratio r:
a_n = a r^(n - 1)
S_n = a (1 - r^n) / (1 - r) r ≠ 1
S_inf = a / (1 - r) only if |r| is less than 1
Harmonic progression (HP): a sequence whose reciprocals form an AP. There is no sum formula. Convert to the AP, work there, and convert back.
Means between a and b:
AM = (a + b)/2
GM = ±sqrt(ab)
HM = 2ab/(a + b)
GM^2 = AM x HM
for positive a ≠ b: AM > GM > HM
Standard sums:
1 + 2 + ... + n = n(n + 1)/2
1^2 + 2^2 + ... + n^2 = n(n + 1)(2n + 1)/6
1^3 + 2^3 + ... + n^3 = [n(n + 1)/2]^2
Recurring decimals: two repeating digits go over 99, three over 999. So 0.(53) repeating is 53/99.
Worked MCQs
Five questions in the style of the practice bank. We checked every answer by substituting back.
1. Find the sum to infinity of (-3/7)^(2n) from n = 1.
An even power kills the minus sign: (-3/7)^(2n) = (9/49)^n. This is a GP with first term 9/49 and ratio 9/49:
S_inf = (9/49) / (1 - 9/49) = (9/49) / (40/49) = 9/40
2. If 2 + 5 + 8 + 11 + ... + a_n = 60100, find n.
AP with a = 2 and d = 3:
(n/2)[4 + 3(n - 1)] = 60100
n(3n + 1) = 120200
Try the options rather than solving the quadratic. n = 200 gives 200 x 601 = 120200. Answer: 200.
3. Write 1.(53), with 53 repeating, as a fraction. Find numerator minus denominator.
1.(53) = 1 + 53/99 = 152/99
152 - 99 = 53
4. A clock strikes twice at 1 o'clock, four times at 2, and so on. How many strikes in 12 hours?
The strikes are 2, 4, 6, ..., 24. That is 2(1 + 2 + ... + 12) = 2 x 78 = 156.
5. In an infinite GP, each term equals twice the sum of all the terms after it. Find r.
Take the first term. The terms after it sum to S - a = a/(1 - r) - a = ar/(1 - r):
a = 2 ar/(1 - r)
1 - r = 2r
r = 1/3
Common traps
- Using
S_infwhen it does not exist. The formula needs the size of r to be less than 1. For r = 2 the series diverges, and an option with a neat number is a trap. - Counting terms. From 2 to 100 in steps of 2 there are 50 terms, not 49. Use
n = (l - a)/d + 1. - GM sign. The geometric mean of 9 and 4 is 6 in most MCQs, but strictly it is ±6. If both 6 and ±6 appear, read the question for "positive".
- Second-order sequences. 2, 5, 9, 14, ... is neither AP nor GP. The differences are 3, 4, 5, so the next differences are 6, 7, 8 and the 7th term is 35. Do not force an AP formula on it.
- HP sums. There is no formula. If a question asks for the sum of an HP, it is almost certainly asking about the corresponding AP.
How to prepare
This is one of the shorter maths chapters and the formulas fit on a card. Learn them, then do a timed set with story questions mixed in, since those cost the most time. The three standard sums come back in the mathematical induction chapter, where you prove them, and the AP sum 1 + 2 + ... + (n - 1) is the same number as nC2 in permutations and combinations.
The formula sheet has all the AP and GP results on one page, and the NET mathematics guide shows where this chapter fits in your plan.
FAQ
What is the difference between AP, GP and HP?
An AP adds the same number each time. A GP multiplies by the same number. An HP is a sequence whose reciprocals form an AP.
When does an infinite geometric series have a sum?
Only when the common ratio lies strictly between -1 and 1. Then the sum is a/(1 - r).
How do I convert a recurring decimal to a fraction quickly?
Put the repeating block over as many 9s as it has digits. 0.(7) is 7/9 and 0.(53) is 53/99. Add any whole-number part afterwards.
Is sequences and series easy for NET?
It is one of the more predictable chapters. Most questions use one of about ten formulas, so a week of practice covers it well.
