Arithmetic and Geometric Progression Notes

A progression is a list of numbers that follows a fixed rule. The two most important types for competitive exams are the Arithmetic Progression (AP), where the same number is added each time, and the Geometric Progression (GP), where each term is multiplied by the same number. Progressions are directly tested in SSC CGL, CDS, NDA and state exams, and they quietly support many other topics: number series in reasoning, savings and instalment problems, compound interest and population growth questions. UPSC CSAT uses AP ideas in counting puzzles, such as the number of seats in rows that increase by a fixed amount or the total distance covered when a runner increases distance daily.

Quick Facts at a Glance

• AP: a, a + d, a + 2d, ... ; nth term = a + (n − 1)d.
• Sum of AP: Sn = n/2 [2a + (n − 1)d] = n/2 (first term + last term).
• GP: a, ar, ar², ... ; nth term = arⁿ⁻¹.
• Sum of GP: Sn = a(rⁿ − 1) ÷ (r − 1), for r ≠ 1.
• Infinite GP (|r| < 1): S∞ = a ÷ (1 − r).
• For positive numbers, AM ≥ GM ≥ HM.

1. Arithmetic Progression (AP)

In an AP, the difference between any two consecutive terms is constant. This constant is called the common difference (d). If d is positive, the AP increases; if negative, it decreases; if zero, all terms are equal.

Examples of AP: 3, 7, 11, 15, ... (d = 4); 50, 45, 40, ... (d = −5).

nth Term of an AP

Tn = a + (n − 1)d, where a is the first term.

Example: Find the 20th term of 5, 9, 13, ... Here a = 5, d = 4. T20 = 5 + 19 × 4 = 81.

Example: Which term of 7, 12, 17, ... is 102? 102 = 7 + (n − 1)5 → n − 1 = 19 → n = 20.

Number of terms in an AP from a to l with difference d = (l − a)/d + 1.

Example: How many two-digit numbers are divisible by 7? From 14 to 98: (98 − 14)/7 + 1 = 13.

Sum of n Terms of an AP

Sn = n/2 [2a + (n − 1)d], or Sn = n/2 (a + l) when the last term l is known.

Example: Sum of the first 25 odd numbers? Odd numbers form an AP with a = 1, d = 2. S25 = 25/2 [2 + 48] = 625. In general, the sum of the first n odd numbers = n².

Example: Sum of all multiples of 3 between 1 and 100: terms 3 to 99, n = 33. Sum = 33/2 × (3 + 99) = 1683.

Properties of an AP

• If a, b, c are in AP, then 2b = a + c; b is the arithmetic mean of a and c.
• Adding, subtracting, multiplying or dividing every term by the same non-zero number keeps it an AP.
• For three terms in AP, take them as a − d, a, a + d. For four terms, take a − 3d, a − d, a + d, a + 3d. This makes the sum easy.
• The sum of terms equidistant from the beginning and end is constant: T1 + Tn = T2 + T(n−1).

Example: Three numbers in AP have sum 27 and product 648. Take them as a − d, a, a + d. Sum = 3a = 27, so a = 9. Product = 9(81 − d²) = 648 → d² = 9 → d = 3. Numbers: 6, 9, 12.

2. Geometric Progression (GP)

In a GP, each term is obtained by multiplying the previous term by a constant called the common ratio (r).

Examples of GP: 2, 6, 18, 54, ... (r = 3); 64, 32, 16, ... (r = 1/2).

nth Term and Sum of a GP

• Tn = arⁿ⁻¹
• Sn = a(rⁿ − 1)/(r − 1) when r > 1, or a(1 − rⁿ)/(1 − r) when r < 1.
• If r = 1, Sn = na.

Example: Find the 8th term of 3, 6, 12, ... T8 = 3 × 2⁷ = 384.

Example: Sum of 1 + 2 + 4 + ... up to 10 terms = (2¹⁰ − 1)/(2 − 1) = 1023.

Infinite GP

When |r| < 1, the terms keep getting smaller and the sum approaches a fixed value: S∞ = a/(1 − r).

Example: 1 + 1/2 + 1/4 + 1/8 + ... = 1/(1 − 1/2) = 2.

Real-life example: A ball is dropped from 10 m and each bounce reaches 3/5 of the previous height. Total distance travelled = 10 + 2 × (10 × 3/5)/(1 − 3/5) = 10 + 2 × 15 = 40 m.

Properties of a GP

• If a, b, c are in GP, then b² = ac; b is the geometric mean of a and c.
• For three terms in GP, take them as a/r, a, ar, so the product is a³.
• Multiplying or dividing every term by the same non-zero number keeps it a GP.

3. Arithmetic, Geometric and Harmonic Means

Mean	Formula for two numbers a and b	Example (4 and 16)
Arithmetic Mean (AM)	(a + b)/2	10
Geometric Mean (GM)	√(ab)	8
Harmonic Mean (HM)	2ab/(a + b)	6.4

For positive numbers, AM ≥ GM ≥ HM, and GM² = AM × HM. Check: 8² = 64 = 10 × 6.4. The harmonic mean is the same formula used for average speed over equal distances.

4. Special Series Sums

Series	Formula	Example
1 + 2 + 3 + ... + n	n(n + 1)/2	Sum to 100 = 5050
1² + 2² + ... + n²	n(n + 1)(2n + 1)/6	Sum to 10 = 385
1³ + 2³ + ... + n³	[n(n + 1)/2]²	Sum to 10 = 3025
Sum of first n odd numbers	n²	First 10 = 100
Sum of first n even numbers	n(n + 1)	First 10 = 110

The story of young Carl Gauss adding 1 to 100 by pairing numbers (1 + 100, 2 + 99, ...) to get 50 × 101 = 5050 is the best way to remember the first formula.

5. AP and GP in Word Problems

Example (savings): Ravi saves ₹500 in the first month and increases his saving by ₹100 every month. Total savings in 12 months = 12/2 [1000 + 11 × 100] = 6 × 2100 = ₹12,600.

Example (growth): A bacteria colony doubles every hour, starting with 50. After 6 hours: 50 × 2⁶ = 3200. This is a GP and is the same idea as compound growth.

Example (CSAT style): A theatre has 20 seats in the first row and each next row has 2 more seats. If there are 15 rows, total seats = 15/2 [40 + 28] = 510.

Common Mistakes to Avoid

1. Using n instead of (n − 1) in the nth-term formula.
2. Confusing AP (constant difference) with GP (constant ratio).
3. Using the infinite GP formula when |r| ≥ 1, where the sum does not settle.
4. Forgetting to count both ends when finding the number of terms.
5. In the bouncing-ball problem, forgetting that each bounce has both an upward and downward distance.

Preparation Strategy

• Learn the six core formulas (nth term and sum for AP, GP and infinite GP) on day one.
• Practise 15 nth-term and number-of-terms questions, then 15 sum questions.
• Memorise the special series sums; they also appear in number system and reasoning series.
• Solve word problems on savings, seats and growth to build the habit of spotting AP and GP in stories.

Practice Questions with Answers

1. Find the sum of the first 20 terms of 2, 7, 12, ... Answer: 10 × (4 + 95) = 990.
2. The 4th term of an AP is 14 and the 10th term is 32. Find d. Answer: 6d = 18, so d = 3.
3. Find the sum of the infinite GP 9 + 3 + 1 + ... Answer: 9 ÷ (1 − 1/3) = 13.5.
4. Find the geometric mean of 2 and 32. Answer: √64 = 8.
5. Find 1² + 2² + ... + 20². Answer: 20 × 21 × 41 ÷ 6 = 2870.

If you can solve these five within five minutes, move on to mixed series questions in the reasoning section.

Related Notes

• Number and Letter Series Notes for Reasoning
• Number System Notes
• Profit, Loss and Interest Notes
• All Quantitative Aptitude Notes

Frequently Asked Questions

What is an arithmetic progression?

An arithmetic progression is a sequence in which the difference between consecutive terms is constant, such as 2, 5, 8, 11.

What is the formula for the nth term of an AP?

Tn = a + (n − 1)d, where a is the first term and d is the common difference.

What is the sum formula of an AP?

Sn = n/2 [2a + (n − 1)d], or n/2 multiplied by the sum of the first and last terms.

What is a geometric progression?

A geometric progression is a sequence in which each term is the previous term multiplied by a constant ratio, such as 3, 6, 12, 24.

What is the formula for the nth term of a GP?

Tn = arⁿ⁻¹, where a is the first term and r is the common ratio.

What is the sum of an infinite GP?

If the common ratio is between −1 and 1, the sum is a ÷ (1 − r).

What is the sum of the first n natural numbers?

n(n + 1) ÷ 2. For example, 1 to 100 adds up to 5050.

What is the relation between AM, GM and HM?

For positive numbers, AM ≥ GM ≥ HM, and GM² = AM × HM for two numbers.

How do you find the number of terms in an AP?

Number of terms = (last term − first term) ÷ common difference + 1.

If a, b, c are in AP, what is the relation?

2b = a + c.

If a, b, c are in GP, what is the relation?

b² = ac.

Where are AP and GP used in exams?

They are used in series questions, savings and instalment problems, growth problems, seat-counting puzzles and compound interest.