AP, GP and Logarithm Formula Guide: Arithmetic Progression, Geometric Progression and Log Laws for SSC
Rahul Kumar
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Progressions and logarithms carry 1 to 3 questions in SSC CGL maths, and they are 100% formula-based. This AP, GP and logarithm formula guide for SSC gives the formulas for the nth term, the sum of the terms, the infinite GP, the laws of logarithms and the standard log values, with solved examples.
Arithmetic Progression (AP)
| Item | Formula |
|---|---|
| nth term | a + (n − 1)d |
| Sum of n terms | n/2 × [2a + (n − 1)d] |
| Sum if the last term l is known | n/2 × (a + l) |
| Common difference | d = any term − the previous term |
| Sum of first n natural numbers | n(n + 1)/2 |
Geometric Progression (GP)
| Item | Formula |
|---|---|
| nth term | a × r^(n − 1) |
| Sum of n terms (r > 1) | a(r^n − 1)/(r − 1) |
| Sum of n terms (r < 1) | a(1 − r^n)/(1 − r) |
| Sum of an infinite GP (|r| < 1) | a/(1 − r) |
| Common ratio | r = any term / the previous term |
Means
| Mean | Formula |
|---|---|
| Arithmetic mean of a and b | (a + b)/2 |
| Geometric mean of a and b | √(ab) |
| Relation | AM ≥ GM |
Laws of Logarithms
| Law | Formula |
|---|---|
| Product | log(mn) = log m + log n |
| Quotient | log(m/n) = log m − log n |
| Power | log(m^n) = n log m |
| Base is the number | log_a a = 1 |
| Log of 1 | log 1 = 0 |
| Change of base | log_a b = log b / log a |
| Relation with powers | If a^x = N, then x = log_a N |
Standard Log Values (Base 10)
| Number | log |
|---|---|
| 2 | 0.3010 |
| 3 | 0.4771 |
| 5 | 0.6990 |
| 7 | 0.8451 |
| 10 | 1 |
Solved Examples
Example 1. Find the 10th term of the AP 2, 5, 8, ...
a = 2, d = 3. 2 + 9 × 3 = 29.
Example 2. Find the sum of the first 20 terms of 3, 5, 7, ...
20/2 × [6 + 19 × 2] = 10 × 44 = 440.
Example 3. Find the 6th term of the GP 3, 6, 12, ...
3 × 25 = 96.
Example 4. Find the sum of the first 5 terms of 2, 4, 8, ...
2(25 − 1)/(2 − 1) = 62.
Example 5. Find the sum of the infinite GP 8, 4, 2, ...
a = 8, r = 1/2. 8/(1 − 1/2) = 16.
Example 6. Find log 6 in terms of log 2 and log 3, and find its value.
log 6 = log 2 + log 3 = 0.3010 + 0.4771 = 0.7781.
Example 7. Find the value of log2 8.
8 = 2³, so the value is 3.
Quick Tips
- Write a, d or r before you use a formula.
- Use n/2 × (a + l) when the last term is known.
- Learn the log values of 2, 3, 5 and 7.
- AM is always greater than or equal to GM.
Quick Revision Points
- AP nth term = a + (n − 1)d.
- GP nth term = ar^(n − 1).
- Infinite GP sum = a/(1 − r).
- log(mn) = log m + log n.
- log 2 = 0.3010 and log 3 = 0.4771.
Practice Questions With Answers
- Find the 20th term of the AP 5, 8, 11, ...
Answer: 62 - Find the sum of the first 50 natural numbers.
Answer: 1,275 - Find the 5th term of the GP 2, 6, 18, ...
Answer: 162 - Find the sum of the infinite GP 6, 3, 1.5, ...
Answer: 12 - Find the value of log2 32.
Answer: 5 - Find the value of log10 1000.
Answer: 3 - Find the value of log 20 (base 10) using log 2 = 0.3010.
Answer: 1.3010 - Find the 10th term of the AP 7, 11, 15, ...
Answer: 43 - Find the AM and the GM of 4 and 16.
Answer: AM = 10 and GM = 8 - What is the value of loga a?
Answer: 1
How to Practise This Chapter
Write each formula on a card, with a solved example on the back, and revise the cards every day. Then test yourself with our practice papers and the previous year papers page. See all guides on the Formula Guide page, and also read the Mathematics notes and Quantitative Aptitude notes. The previous guide is SI Units and Physical Quantities and the next guide is Clock and Calendar. Also read Arithmetic and Geometric Progression.
Frequently Asked Questions
What is the nth term of an AP?
a + (n − 1)d.
What is the sum of n terms of an AP?
n/2 times [2a + (n − 1)d].
What is the nth term of a GP?
a times r to the power (n − 1).
What is the sum of an infinite GP?
a/(1 − r), if the absolute value of r is less than 1.
What is the log of a product?
The sum of the logs: log(mn) = log m + log n.
What is the value of log 1?
It is 0.
What is log base a of a?
It is 1.
What is log 2 in base 10?
About 0.3010.
What is AM and GM?
AM is the arithmetic mean (a + b)/2 and GM is the geometric mean √(ab).
Is this topic asked in SSC exams?
Yes. AP, GP and logarithm questions are asked in SSC CGL maths.
How much time should I take?
About 40 seconds.
What are common mistakes?
Using the wrong n and mixing up d and r.
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SI Units and Physical Quantities List: Base Units, Derived Units, Conversions and Scientists for SSC
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Clock and Calendar Formula Guide: Angle, Overlap, Odd Days and Day Finding Formulas for SSC Reasoning
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.