Paper I — Mathematics

Sequences and Series (AP, GP, HP)

Exam-focused notes, visual explanation, traps, revision sheet and every mapped past NDA question.

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Core5 mapped NDA questions

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Build exam-ready understanding by connecting the core idea, formulas or facts, worked example and previous NDA questions. Focus on why each rule works, then test it under timed conditions.

Concept

An arithmetic progression (AP) has a constant common difference d; a geometric progression (GP) has a constant common ratio r; a harmonic progression (HP) is the reciprocal of an AP. These appear in nearly every paper and reward quick formula recall.

The arithmetic, geometric and harmonic means satisfy AM ≥ GM ≥ HM for positive numbers.

Key Points & Formulas

AP: nth term aₙ = a + (n−1)d; sum Sₙ = n/2 [2a + (n−1)d].
GP: nth term aₙ = a r^(n−1); sum Sₙ = a(rⁿ−1)/(r−1), r≠1.
Infinite GP (|r|<1): S∞ = a/(1−r).
For two numbers: AM = (a+b)/2, GM = √(ab), and GM² = AM·HM.

Worked Example

Q. The 4th term of a GP is 24 and the 7th term is 192. Find the common ratio.

a r⁶ / a r³ = r³ = 192/24 = 8.

r = ∛8 = 2.

Common traps

Reading the prompt too quickly and missing a qualifier.
Memorising the result without checking when it applies.
Skipping units, signs, chronology or the final reasonableness check.

NDA shortcuts

State the governing rule before substituting values.
Eliminate impossible options before doing full calculation.

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5 NDA questions

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Question 1

NDA I 2026

If $p^\text{th}$ term of an AP is $k$ , then what is the sum of $p^\text{th}$ term, $(p+q)^\text{th}$ term and $(p-q)^\text{th}$ term of the AP?

Question 2

NDA II 2025

If $(10 + \log_{10} x)$ , $(10 + \log_{10} y)$ and $(10 + \log_{10} z)$ are in AP, then consider the following statements:

I. The GM of $x$ and $z$ is $y^2$ .

II. The AM of $\log_{10} x$ and $\log_{10} z$ is $\log_{10} y$ .

Which of the statements given above is/are correct?

Question 3

NDA II 2024

Let x > 1, y > 1, z > 1 be in GP. Then 1/(1 + ln x), 1/(1 + ln y), 1/(1 + ln z) are

Question 4

NDA II 2024

What is the common difference?

Question 5

NDA II 2024

What is the sum of all five terms?

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