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CMA Foundation · Fundamentals of Business Mathematics and Statistics

Arithmetic Progression and Geometric Progression for CMA Foundation

An arithmetic progression (AP) adds a fixed number each step; a geometric progression (GP) multiplies by a fixed number each step. To solve MCQs, identify the type, find d or r, then apply Tn = a + (n − 1)d or Tn = arⁿ⁻¹ and the matching sum formula.

What this chapter covers

This chapter covers two kinds of number patterns. In an AP, the difference between consecutive terms is constant. In a GP, the ratio between consecutive terms is constant. You learn the nth term, the sum of n terms, and for a GP the sum to infinity. You also learn the arithmetic mean, geometric mean and the link between AM, GM and HM.

The chapter is formula-driven and short on theory. Most questions need two or three lines of working. If you know which formula to use and can handle simple algebra, you can answer each one in under a minute.

It connects to other parts of Paper 3. Simple and compound interest, annuities and depreciation use the same logic as AP and GP. Averages in Statistics use AM, GM and HM again. Practising this chapter also improves your speed with powers and fractions, which helps elsewhere in the paper.

Paper 3 is fully objective, with 50 MCQs in one hour and no negative marking. AP and GP questions are formula-based and have one clear answer, so they are among the most reliable marks in Mathematics if you prepare well. The chapter is small, the formulas are few, and a week of focused practice is enough to answer almost any question in 30 to 60 seconds. That saves time for harder questions in the same paper. Since there is no negative marking, you should attempt every question, and a quick elimination often gets you to the right option even when you are unsure.

Arithmetic Progression and Geometric Progression: topics in the order to study them

  1. 1Sequences and Series BasicsStart here to understand what a term, a sequence and a series are, since every later formula uses this language.
  2. 2Arithmetic Progression: nth Term and Common DifferenceAP is the simpler pattern, so learn Tn = a + (n − 1)d and how to find d before moving on.
  3. 3Sum of n Terms of an AP and Arithmetic MeanThe sum formula builds directly on the nth term, and the arithmetic mean of two numbers is the middle term of an AP.
  4. 4Geometric Progression: nth Term and Common RatioOnce AP is solid, compare it with GP and learn Tn = arⁿ⁻¹ and how to find r.
  5. 5Sum of GP and Sum to InfinityThe sum formulas need the nth term and the ratio, and the infinite sum applies only when |r| < 1.
  6. 6Geometric Mean and Relation between AM, GM and HMThis ties both progressions to averages and gives the relation GM² = AM × HM for two positive numbers.
  7. 7Applications of AP and GP in BusinessStudy this last, because word problems on savings, growth and depreciation need all the earlier formulas.

How to prepare Arithmetic Progression and Geometric Progression

Treat this chapter as a formula toolkit. Understand each formula once, then build speed through short, timed practice.

  1. Write the formulas on one page: Tn = a + (n − 1)d, Sn = n/2 × [2a + (n − 1)d], Tn = arⁿ⁻¹, Sn = a(rⁿ − 1) ÷ (r − 1), and S∞ = a ÷ (1 − r).
  2. For each question, first decide whether the pattern is AP or GP by checking the difference and the ratio of the first three terms.
  3. Practise finding a, d, r and n from given information, because many MCQs hide the data in words such as 'the 5th term is 20'.
  4. Solve two simultaneous equations from two given terms, for example T3 and T7 of an AP, until it feels routine.
  5. Do small numeric examples of the AM, GM and HM relation, and check that AM ≥ GM ≥ HM for positive numbers.
  6. Work through business word problems and translate each into a, d or r before using any formula.
  7. Finish with a timed set of 20 mixed MCQs and aim for under 45 seconds each. Review every wrong answer for the cause.

Common mistakes in Arithmetic Progression and Geometric Progression

  • Using n instead of n − 1 in the nth term formula.

    Fix: Test the formula with n = 1. It must give the first term a. If it does not, you have the wrong power or multiplier.

  • Using the sum to infinity when |r| is not less than 1.

    Fix: Check |r| < 1 first. If r is 2 or −1.5, the infinite sum does not exist.

  • Confusing the common difference with the common ratio.

    Fix: Check both on the first three terms. Equal differences mean AP; equal ratios mean GP.

  • Taking the wrong sign when the terms decrease.

    Fix: Always compute d as later term minus earlier term. Here d = −3.

  • Counting the number of terms wrongly, for example from 5 to 50 with d = 5.

    Fix: Use n = (l − a) ÷ d + 1. Here n = 45 ÷ 5 + 1 = 10.

  • Mixing up the AM, GM and HM relation.

    Fix: Remember it holds for positive numbers, AM ≥ GM ≥ HM, and for two numbers GM² = AM × HM. Check with 4 and 16: AM = 10, GM = 8, HM = 6.4, and 8² = 64 = 10 × 6.4.

Last-day revision: Arithmetic Progression and Geometric Progression

  • AP: each term is the previous term plus a constant d; d = T2 − T1.
  • nth term of an AP: Tn = a + (n − 1)d.
  • Sum of n terms of an AP: Sn = n/2 × [2a + (n − 1)d] = n/2 × (first term + last term).
  • AM of two numbers a and b is (a + b) ÷ 2, and it is the middle term of an AP with those ends.
  • GP: each term is the previous term times a constant r; r = T2 ÷ T1.
  • nth term of a GP: Tn = arⁿ⁻¹.
  • Sum of n terms of a GP: Sn = a(rⁿ − 1) ÷ (r − 1) for r ≠ 1; if r < 1 you may use a(1 − rⁿ) ÷ (1 − r).
  • Sum to infinity of a GP: S∞ = a ÷ (1 − r), valid only when |r| < 1.
  • GM of two positive numbers a and b is √(ab).
  • For two positive numbers: AM ≥ GM ≥ HM, and GM² = AM × HM.
  • Three numbers in AP can be taken as a − d, a, a + d; in GP as a/r, a, ar.
  • Number of terms from first to last in an AP: n = (l − a) ÷ d + 1.

Arithmetic Progression and Geometric Progression practice questions

Arithmetic Progression and Geometric Progression in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Arithmetic Progression and Geometric Progression: frequently asked questions

How do I tell whether a series is an AP or a GP in an MCQ?

Take the first three terms. If T2 − T1 equals T3 − T2, it is an AP. If T2 ÷ T1 equals T3 ÷ T2, it is a GP. Do this in a few seconds before choosing a formula.

Do I need to memorise the derivations of the AP and GP formulas?

No. The paper is fully objective, so you only need to know the formulas, their conditions and how to apply them quickly. Understanding where a formula comes from helps you remember it, but the proof is not asked.

When can I use the sum to infinity of a GP?

Only when the common ratio lies strictly between −1 and 1, that is |r| < 1. In that case the sum is a ÷ (1 − r). If |r| is 1 or more, the terms do not shrink and the sum does not settle on a value.

How many AP and GP questions should I expect in the exam?

The number can vary from one exam to the next, so do not rely on a fixed count. Prepare the whole chapter, since its formulas are short and the questions are quick to solve.