Letter, Symbol and Number Series notes

Basic Method for Solving Series

A series is an ordered group of letters, numbers, symbols, or mixed terms. The missing term is found by identifying the rule that changes one term into the next.

For letters, use their positions in the English alphabet: A = 1, B = 2, ..., Z = 26. For numbers, calculate differences, ratios, products, or alternating operations. Always check the rule across all available terms before selecting an answer.

• Alphabet positions are A1, B2, C3, D4, E5, F6, G7, H8, I9, J10, K11, L12, M13, N14, O15, P16, Q17, R18, S19, T20, U21, V22, W23, X24, Y25, Z26.
• In a cyclic alphabet series, after Z counting begins again from A. For example, U, B, I, P, W, D increases by 7 positions each time when the alphabet is treated cyclically.
• Common patterns include constant addition, constant subtraction, increasing differences, multiplication, division, alternating operations, and two interwoven series.
• If a series has letters and numbers together, solve the letter pattern and number pattern separately.
• A valid rule should normally explain every given term, not just the first two terms.

Alphabet Series and Letter Positions

In a simple alphabet series, the letters move forward or backward by a fixed number of positions. In a more difficult series, the jumps increase, decrease, or follow separate patterns.

For example, B, E, J, Q, Z has positions 2, 5, 10, 17, 26. The differences are +3, +5, +7, and +9. Therefore, each jump increases by 2.

Some series contain groups of letters. Convert every letter in each group into its alphabet position and compare the corresponding positions.

• B, E, J, Q, Z follows positions 2, 5, 10, 17, 26, with differences +3, +5, +7, +9.
• U, B, I, P, W, D follows a cyclic jump of +7: U to B, B to I, I to P, P to W, and W to D.
• A series such as AZ, CX, EV, GT has first letters increasing by 2 and second letters decreasing by 2.
• AC, BF, DJ, GO, KU has first-letter positions A, B, D, G, K, with jumps +1, +2, +3, +4. Its second letters C, F, J, O, U have jumps +3, +4, +5, +6.
• ACD, AGD, AJD, AND keeps the first and third letters fixed. The changing middle letters are C, G, J, N.
• The series CDEF, DFHJ, EHKN, FJNR, GLQV changes each corresponding letter by increasing jumps. The jumps between terms are +1, +2, +3, +4, and +5.

Mixed Letter and Number Series

In a mixed series, letters and numbers usually follow independent rules. First write the alphabet position of each letter. Then inspect the number part separately.

For B4, E8, I14, N22, the letter positions are 2, 5, 9, 14. Their differences are +3, +4, +5, so the next letter is T, position 20. The numbers are 4, 8, 14, 22, with differences +4, +6, +8, so the next number is 32.

• B4, E8, I14, N22, T32 uses increasing letter jumps of +3, +4, +5, +6.
• A1, C4, F9, J16, O25 has letters with jumps +2, +3, +4, +5 and numbers equal to the squares 1, 4, 9, 16, 25.
• W-144, U-121, S-100, Q-81, O-64 has letters decreasing by 2 and numbers equal to the squares of 11, 10, 9, 8, and 7.
• When a term is written as letter-number, do not assume the number is the alphabet position. It may be a square, multiple, or another independent pattern.
• For B4, E8, I14, N22, the next term is T32 because T is the twentieth letter and the next number after 22 is 32.
• A letter pattern can move forward while the number pattern moves backward, as in W-144 to U-121.

Number Series: Differences and Operations

The first step in a number series is to find the difference between consecutive terms. If the differences are not constant, calculate second differences or look for multiplication and addition together.

Some series use an alternating rule. Separate odd-position terms and even-position terms if one operation does not explain the whole sequence.

• 12, 32, 72, 152, 312 follows multiplication by 2 and addition by 8: 12 x 2 + 8 = 32, 32 x 2 + 8 = 72. The next term is 312.
• 2, 5, 15, 18, 54, 57, 171 alternates +3 and x3: 2 + 3 = 5, 5 x 3 = 15. The next term is 171.
• 5, 10, 8, 16, 14, 28, 26, 52 alternates x2 and -2. The next term after 26 is 52.
• 6, 13, 28, 59, 122 follows x2 + 1, x2 + 2, x2 + 3, and x2 + 4.
• 6, 11, 21, 36, 56, 81 has differences 5, 10, 15, 20, and 25. The next term is 81.
• 2, 6, 7, 13, 14, 22, 23, 33 can be viewed as two interwoven series: odd-position terms 2, 7, 14, 23 increase by 5, 7, 9, while even-position terms 6, 13, 22, 33 increase by 7, 9, 11.
• 1, 5, 13, 25, 41 has differences 4, 8, 12, and 16. The next term is 41.

Special Number Patterns and Wrong Terms

Many questions use familiar patterns such as consecutive squares, repeated additions, alternating terms, or numbers formed by joining digits. Write the expected rule clearly before choosing the missing term.

For a wrong-term question, calculate the expected pattern from the beginning. The incorrect term is the one that breaks the established rule.

• 8, 10, 14, 18, 26, 34, 50, 66 has repeated differences: +2, +4, +4, +8, +8, +16, +16. The missing term is 26.
• 80, 10, 70, 15, 60, 20 alternates two decreasing and increasing series. The first group decreases by 10, while the second group increases by 5.
• 3, 9, 15, ? is intended in the supplied series to give 33. Since only three terms are shown, more than one mathematical rule can fit them, so the intended pattern must be followed in an examination question.
• 234, 2345, 23456, 234567, 2345678 adds the next digit to the right each time. The missing final digit is 8.
• In 1, 2, 5, 10, 17, 28, the expected rule is adding consecutive even numbers: +2, +4, +6, +8, +10. Therefore, 28 is the wrong term because the expected final term is 27.
• A series based on squares can be recognised from 1, 4, 9, 16, 25, and so on.
• When the terms are too few to prove one rule, compare the answer choices and select the rule that is consistent with the intended sequence.

Letter Pair and Common-Letter Questions

Letter-pair questions compare the distance between two letters in a word with the distance between the same letters in the alphabet. Count the spaces or letters carefully. The wording usually asks for letters between, not the position difference itself.

Common-letter questions require matching distinct letters in two words. Repeated letters are generally counted according to the wording. In the example Quote and Poet share O, T, and E.

• For two letters in a word, count the number of letters lying between their positions in the word.
• For the alphabet, count the number of letters lying between their alphabet positions.
• In HISTORICAL, the pairs satisfying the condition are H-I, I-C, S-T, S-L, T-R, and I-L. Hence, there are six such pairs.
• The two occurrences of I in HISTORICAL must be treated as separate positions when checking pairs.
• Quote contains Q, U, O, T, E. Poet contains P, O, E, T. Their common letters are O, T, and E, so the answer is 3.
• Do not count letters that merely look similar. Compare the actual alphabet characters.
• For pair questions, compare all possible pairs, not only adjacent letters.

Counting Rectangles and Squares

Figure-counting questions require every possible rectangle to be counted, including large rectangles made by combining smaller regions. A square is also a rectangle, but questions may ask for rectangles and squares separately or together.

Begin by counting small shapes. Then count shapes covering two or more adjacent regions. Finally count the complete outer figure. Avoid counting the same boundary twice.

• A square has four equal sides and four right angles.
• A rectangle has four right angles, but its adjacent sides do not have to be equal.
• Every square is a rectangle, but every rectangle is not a square.
• For an m by n grid of small rectangles, the total number of rectangles is m(m + 1)n(n + 1) divided by 4.
• For a grid, count squares separately by size, such as 1 by 1, 2 by 2, and 3 by 3, then add them if the question asks for both types.
• Use the lines as boundaries. A shape counts only if all four sides can be traced without interruption.
• In the supplied figure-counting question, the total count of rectangles and squares is 15.

Practical Exam Procedure and Answer Checking

Series questions can often be solved quickly by writing alphabet positions or first differences. Do not apply multiplication when simple differences already explain the terms. If the direct pattern fails, test alternating positions or separate components.

After obtaining an answer, substitute it back into the series. This confirms whether the rule continues correctly.

• For letter-only series, write numerical alphabet positions beside the letters.
• For number series, calculate first differences before testing products or ratios.
• For alternating patterns, inspect terms 1, 3, 5 and terms 2, 4, 6 separately.
• For alphanumeric terms, solve the letter and number components independently.
• For word questions, write the letters of both words and mark the common letters.
• For figure questions, count small, medium, large, and complete shapes.
• Check the final answer by applying the same rule from the previous term to the selected term.