Arithmetic and Geometric Progression Notes for SSC Exams
Rahul Kumar
SSC Exam Expert & Content Editor
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A sequence is a list of numbers in a fixed pattern, and the two most important patterns are the arithmetic progression (AP) and the geometric progression (GP). SSC exams ask the nth term, the sum of terms and the number of terms in these progressions. These AP and GP notes for SSC explain every formula with solved examples and practice questions.
Arithmetic Progression (AP)
In an AP, the difference between two consecutive terms is always the same. This is called the common difference (d). Example: 3, 7, 11, 15, ... has d = 4.
| Item | Formula |
|---|---|
| nth term | an = a + (n − 1) d |
| Sum of n terms | Sn = (n / 2) [2a + (n − 1) d] |
| Sum using the last term | Sn = (n / 2) (a + l) |
| Number of terms | n = (l − a) / d + 1 |
| Common difference | d = a2 − a1 |
| Arithmetic mean of a and b | (a + b) / 2 |
| Condition for a, b, c in AP | 2b = a + c |
Sums of Standard Series
| Series | Sum of n terms |
|---|---|
| 1 + 2 + 3 + ... + n | n (n + 1) / 2 |
| 1 + 3 + 5 + ... (n odd numbers) | n2 |
| 2 + 4 + 6 + ... (n even numbers) | n (n + 1) |
| 12 + 22 + ... + n2 | n (n + 1) (2n + 1) / 6 |
| 13 + 23 + ... + n3 | [n (n + 1) / 2]2 |
Solved Examples on AP
Example 1. Find the 5th term of the AP 3, 7, 11, ...
a = 3, d = 4, so a5 = 3 + 4 × 4 = 19.
Example 2. Find the sum of the first 10 terms of 2, 5, 8, ...
S10 = (10 / 2) [2 × 2 + 9 × 3] = 5 × 31 = 155.
Example 3. How many terms are there in the AP 4, 9, 14, ..., 84?
n = (84 − 4) / 5 + 1 = 16 + 1 = 17.
Geometric Progression (GP)
In a GP, each term is obtained by multiplying the previous term by the same number, called the common ratio (r). Example: 2, 6, 18, 54, ... has r = 3.
| Item | Formula |
|---|---|
| nth term | an = a rn − 1 |
| Sum of n terms (r > 1) | Sn = a (rn − 1) / (r − 1) |
| Sum of n terms (r < 1) | Sn = a (1 − rn) / (1 − r) |
| Sum to infinity (|r| < 1) | S = a / (1 − r) |
| Common ratio | r = a2 / a1 |
| Geometric mean of a and b | √(ab) |
| Condition for a, b, c in GP | b2 = a c |
Solved Examples on GP
Example 4. Find the 5th term of the GP 2, 6, 18, ...
a = 2, r = 3, so a5 = 2 × 34 = 162.
Example 5. Find the sum of the first 5 terms of 2, 6, 18, ...
S5 = 2 (35 − 1) / (3 − 1) = 2 × 242 / 2 = 242.
Example 6. Find the sum of the infinite GP 1/2 + 1/4 + 1/8 + ...
a = 1/2 and r = 1/2, so S = (1/2) / (1 − 1/2) = 1.
Difference Between AP and GP
| Point | AP | GP |
|---|---|---|
| Pattern | Add the same number each time | Multiply by the same number each time |
| Constant | Common difference d | Common ratio r |
| Growth | Steady (linear) | Fast (exponential) |
| Example | 5, 8, 11, 14 | 5, 10, 20, 40 |
Quick Tips
- Find a and d (or r) first, and then use the formula.
- For the number of terms, use n = (l − a) / d + 1.
- Use the standard sums of natural, odd and even numbers directly.
- For an infinite GP, check that the ratio lies between −1 and 1.
Quick Revision Points
- AP: an = a + (n − 1) d.
- AP sum: Sn = (n / 2) (a + l).
- GP: an = a rn − 1.
- Infinite GP sum: a / (1 − r), if |r| < 1.
- Sum of first n natural numbers = n (n + 1) / 2.
Practice Questions With Answers
- Find the 10th term of the AP 5, 8, 11, ...
Answer: 32 - Find the sum of the first 20 natural numbers.
Answer: 210 - Find the sum of the first 10 odd numbers.
Answer: 100 - Find the number of terms in 7, 11, 15, ..., 99.
Answer: 24 - Find the 6th term of the GP 3, 6, 12, ...
Answer: 96 - Find the common ratio of the GP 4, 12, 36, ...
Answer: 3 - Find the sum to infinity of 9 + 3 + 1 + ...
Answer: 13.5 - If 3, x, 27 are in GP, find x (positive).
Answer: 9 - If 5, x, 21 are in AP, find x.
Answer: 13 - Find the sum of the first 5 terms of 1, 2, 4, 8, ...
Answer: 31
How to Practise This Chapter
Solve five AP and GP problems daily and write the formula before each one. Then attempt timed sets from our practice papers and the previous year papers page. Read the previous chapter, Coordinate Geometry, and the next chapter, Linear Equations in Two Variables. See all topics on the Mathematics notes page.
Frequently Asked Questions
What is an arithmetic progression?
An arithmetic progression is a sequence in which the difference between consecutive terms is always the same.
What is the formula of the nth term of an AP?
The nth term is a plus (n minus 1) times d, where a is the first term and d is the common difference.
What is the formula of the sum of an AP?
The sum of n terms is n by 2 times (2a plus (n minus 1) d), or n by 2 times (first term plus last term).
What is a geometric progression?
A geometric progression is a sequence in which each term is the previous term multiplied by a fixed common ratio.
What is the formula of the nth term of a GP?
The nth term is a times r to the power (n minus 1).
What is the sum of an infinite GP?
The sum is a divided by (1 minus r), and it exists only if the absolute value of r is less than 1.
How do I find the number of terms of an AP?
Number of terms is (last term minus first term) divided by d, plus 1.
What is the sum of the first n natural numbers?
It is n times (n plus 1) divided by 2.
What is the condition for three numbers to be in AP or GP?
Three numbers a, b, c are in AP if 2b equals a plus c, and in GP if b squared equals a times c.
What is the difference between AP and GP?
In an AP you add the same number each time, and in a GP you multiply by the same number each time.
Is this topic important for SSC?
Yes. Sequence and series questions are asked in SSC CGL and in the higher tiers.
How many questions come from AP and GP in SSC exams?
Usually one or two questions come from this topic, but the number changes from year to year.
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Rahul Kumar
SSC Exam Expert & Content Editor
Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.