Number and Alphabet Series
Number and Alphabet Series is a very common topic of the reasoning paper of ADRE Grade IV. In these questions a series of numbers or letters is given in which one term is missing or wrong, and the candidate has to find the term that continues or corrects the series. The topic tests the ability to notice a pattern. The pattern may be based on addition, subtraction, multiplication, squares, cubes, or the position of letters in the alphabet. With regular practice, a candidate learns to recognise the common patterns almost at a glance.
What a Series Is
A series is a list of numbers or letters arranged in a fixed order according to a rule. Each item of the series is called a term. The rule that joins one term to the next is called the pattern. In a number series the pattern is usually an arithmetic change, such as adding a fixed number, or a growing change, such as adding increasing numbers. In an alphabet series the pattern is the movement of letters forward or backward in the alphabet, or the skipping of a fixed number of letters between terms.
Common Number Patterns
- Constant difference — 2, 5, 8, 11 (add 3 each time).
- Constant ratio — 3, 6, 12, 24 (multiply by 2 each time).
- Increasing difference — 1, 2, 4, 7, 11 (add 1, 2, 3, 4).
- Squares — 1, 4, 9, 16, 25 (squares of natural numbers).
- Cubes — 1, 8, 27, 64, 125 (cubes of natural numbers).
- Alternating series — 2, 10, 4, 20, 6 (two patterns mixed).
- Prime numbers — 2, 3, 5, 7, 11, 13.
Common Alphabet Patterns
In an alphabet series, first write the letters and number them from one to twenty-six. For example, in the series A, C, E, G the letters move forward by two places each time. In the series Z, X, V, T the letters move backward by two places. Sometimes the pattern uses the gap between the letters, as in A, D, G, J, where two letters are skipped each time. When two interleaved series are combined, such as A, B, D, E, H, I, one series is A, D, H and the other is B, E, I.
Method of Solving a Series
- Write the given terms in a row and look at the change from one term to the next.
- Find whether the change is by addition, subtraction, multiplication or division.
- If the change is not constant, see whether it is increasing or decreasing.
- Test squares, cubes and prime numbers if no simple rule is found.
- For letter series, write the alphabet with position numbers beside you.
Worked Example
Consider the series 2, 6, 12, 20, ?. The differences are 4, 6 and 8, which increase by two each time. The next difference should be 10, so the next term is 20 + 10 = 30. As another example, consider A, C, F, J, ?. Here the gaps are one, two and three letters, so the next gap should be four letters, which gives the letter O. These examples show how the difference between terms often reveals the rule.
How This Topic Appears in the Examination
Number and alphabet series usually appears as three to five objective questions of one mark each. Most questions give a series with one missing term, and a few ask for the wrong term in a series. The questions are short and can be solved in less than a minute once the pattern is found. A candidate who has practised the common patterns recognises them quickly, and the topic then becomes one of the easiest sources of marks in the reasoning paper.
Memory Tips
Always find the difference between consecutive terms before doing anything else, since most number series are based on the difference. Learn the squares up to thirty and the cubes up to fifteen by heart. For letter series, keep the alphabet with position numbers in front of you. When a series looks complicated, check whether two simple series have been mixed together. Write the terms in a row with enough space so that the differences can be noted below them.
Common Mistakes to Avoid
- Looking at the whole series at once instead of the change between terms.
- Stopping after finding one rule without checking that it fits every term.
- Forgetting the increasing or decreasing nature of the difference.
- Making an error in the position of letters in the alphabet.
- Confusing a square series with a cube series.
How to Prepare This Topic
- Learn the squares and cubes of numbers by heart.
- Practise finding the difference between terms in every series.
- Solve at least fifteen series questions every day.
- Keep a list of the different patterns you meet.
- For letters, revise the alphabet positions daily.
More Solved Examples
In the series 2, 6, 12, 20, 30, the differences are 4, 6, 8, 10, so the next difference is 12 and the next term is 42. In 3, 9, 27, 81 each term is multiplied by 3, so the next term is 243. In the letter series A, C, F, J, O, the gaps are 2, 3, 4, 5 letters, so the next gap is 6 and the next letter is O + 6 = U. When two series are mixed, as in A1, C3, E5, G7, separate the letter part and the number part and solve each one on its own, which makes an otherwise difficult question quite simple.
Advanced Tips for the Exam
Write the differences between consecutive terms below the series, because the pattern often appears in the differences rather than in the terms themselves. Test multiplication and division when the jump between terms is large. In letter series, convert the letters into numbers (A = 1) so that the pattern becomes an ordinary number series. Check whether two alternating sequences are hidden inside one series. Always verify the pattern on the last two or three terms before choosing the answer, since this removes careless mistakes.
Quick Revision Checklist
- Squares and cubes of numbers up to twenty.
- Prime numbers up to fifty.
- Common difference and common ratio patterns.
- Alphabet positions from A = 1 to Z = 26.
- Mixed series split into two parts.
- Verifying the pattern on the last terms.
Frequently Asked Questions
What is the first thing to do in a number series?
Always find the difference between consecutive terms first. If the difference is constant or increasing in a fixed way, that itself gives the rule of the series.
How do I solve an alphabet series?
Write the alphabet with the letters numbered from one to twenty-six, then note how many places each letter moves forward or backward. Skipped letters are counted in the same way.
What are the common number series patterns?
The common patterns are constant difference, constant ratio, increasing difference, squares, cubes, prime numbers, and two simple series mixed together alternately.
What should I do if no simple rule fits?
Test the squares and cubes of natural numbers, the prime numbers, and the possibility that two separate series have been combined. Often one of these reveals the pattern.
How much practice is needed for this topic?
Regular practice of about fifteen questions a day for a few weeks is enough. The patterns are limited, so once they are learnt the questions become quick and reliable.