Letter, Number & Symbol Series
Arithmetic Number Series
An arithmetic sequence is a series of numbers where each term increases or decreases by a fixed amount called the common difference ($d$). Recognising the constant difference is the most direct way to decode a number series.
$$a_n = a_1 + (n - 1)d$$
Gives the value of the nth term in an arithmetic sequence when you know the first term and the common difference.
$a_n$=nth term of the sequence(same as the terms)
$a_1$=first term(same as the terms)
$n$=position number of the term(dimensionless)
$d$=common difference between consecutive terms(same as the terms)
$d = 0$
→Every term equals $a_1$ — a constant sequence
$d < 0$
→The sequence decreases (e.g. 20, 17, 14, …)
Constant difference: Subtract any two consecutive terms to find $d$. If the difference is the same everywhere, the series is arithmetic.
Positive $d$: The series rises (e.g. 4, 9, 14, 19, … has $d = 5$).
Negative $d$: The series falls (e.g. 50, 43, 36, 29, … has $d = -7$).
Sometimes a series question asks for the sum of terms rather than a single term. The sum of a finite arithmetic sequence can be found without adding every term individually.
$$S_n = \frac{n}{2}\,(a_1 + a_n)$$
Computes the sum of the first n terms of an arithmetic sequence by averaging the first and last term, then multiplying by the number of terms.
$S_n$=sum of the first n terms(same as the terms)
$n$=total number of terms(dimensionless)
$a_1$=first term(same as the terms)
$a_n$=nth (last) term(same as the terms)
$a_1 = a_n$
→All terms are equal, so $S_n = n \cdot a_1$
Pairing insight: The first and last terms add up to the same sum as the second and second-last, and so on — there are $n/2$ such pairs.
When $n$ is odd: The formula still works because $\frac{n}{2}(a_1 + a_n)$ correctly accounts for the unpaired middle term.
Geometric Progression
A geometric sequence is a series where each term is obtained by multiplying the previous term by a fixed number called the common ratio ($r$). Unlike an arithmetic sequence, the change between terms grows larger (or smaller) over time.
$$a_n = a_1 \cdot r^{\,n-1}$$
Gives the value of the nth term of a geometric sequence using the first term and the common ratio.
$a_n$=nth term of the sequence(same as the terms)
$a_1$=first term(same as the terms)
$r$=common ratio — each term divided by the previous term(dimensionless)
$n$=position number of the term(dimensionless)
$r = 1$
→Every term equals $a_1$ — a constant sequence
$0 < r < 1$
→Terms shrink toward zero (e.g. 64, 32, 16, … with $r = 0.5$)
$r < 0$
→Terms alternate in sign (e.g. 3, −6, 12, −24, … with $r = -2$)
Finding $r$: Divide any term by the one before it: $r = a_2 / a_1$. Check at least two pairs to confirm the ratio is constant.
Growth behaviour: When $|r| > 1$ the terms grow rapidly; when $|r| < 1$ they shrink.
Fractional $r$: A common ratio can be a fraction, e.g. 81, 27, 9, 3, … has $r = \frac{1}{3}$.
The sum of a geometric series adds all terms up to a given position. This is useful when a question presents several terms and asks for their total.
$$S_n = a_1 \cdot \frac{r^{\,n} - 1}{r - 1} \quad (r \neq 1)$$
Computes the sum of the first n terms of a geometric sequence with common ratio r.
$S_n$=sum of the first n terms(same as the terms)
$a_1$=first term(same as the terms)
$r$=common ratio(dimensionless)
$n$=number of terms(dimensionless)
$r = 1$
→Formula is undefined; use $S_n = n \cdot a_1$ instead
$0 < r < 1$
→Rewrite as $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$ to avoid negative numerators
Derivation shortcut: Multiply $S_n$ by $r$, subtract from $S_n$, and most terms cancel — leaving the compact formula.
When $r < 1$: It is cleaner to write $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$ so the numerator stays positive.
Multi-Step and Mixed Number Patterns
Not every number series follows a single constant difference or ratio. A two-step pattern applies one arithmetic rule to produce the differences, then a second rule to produce the differences-of-differences. Detecting these layered rules requires computing successive differences between terms.
First-level differences: Subtract each term from the next. If these differences are not constant, compute second-level differences.
Second-level differences: Subtract each first-level difference from the next. If these are constant, the original series follows a quadratic pattern.
Alternating series: The series may alternate between two different rules on odd and even positions (e.g. +3, ×2, +3, ×2, …).
Layered Difference Example: 2, 5, 10, 17, 26, ?
1
Original series: 2, 5, 10, 17, 26
2
First-level differences: 3, 5, 7, 9
3
Second-level differences: 2, 2, 2 (constant!)
4
Next first-level difference: 9 + 2 = 11
5
Next term: 26 + 11 = 37
Series can also be built from square numbers, cube numbers, Fibonacci-style sums, or other mathematical constructs. Recognising these special patterns speeds up identification significantly.
Square numbers: 1, 4, 9, 16, 25, … — each term is $n^2$ where $n = 1, 2, 3, \ldots$
Cube numbers: 1, 8, 27, 64, 125, … — each term is $n^3$.
Fibonacci-style: Each term is the sum of the two preceding terms (e.g. 2, 3, 5, 8, 13, …).
Product pattern: Consecutive products like $1 \times 2, 2 \times 3, 3 \times 4, \ldots$ give 2, 6, 12, 20, 30, …
Quick Pattern Recognition Guide
•
Differences are constant → Arithmetic sequence
•
Ratios are constant → Geometric sequence
•
Differences form an arithmetic sequence → Quadratic (two-step)
•
Each term = sum of previous two → Fibonacci-style
•
Terms are perfect squares or cubes → Check $n^2$ or $n^3$
•
Terms grow very fast → Could be factorial ($n!$) or exponential
Letter Series and Alphabet Position
A letter series treats each letter as its position in the alphabet: A = 1, B = 2, …, Z = 26. The core strategy is to convert letters to numbers, find the pattern rule, and convert back.
Alphabet position: A through Z map to 1 through 26. This conversion turns a letter series into a number series.
Reverse position: Z = 1, Y = 2, …, A = 26. Some patterns use the mirror position — always note if the letters are moving backwards through the alphabet.
Vowel/consonant distinction: Some series classify letters as vowels (A, E, I, O, U) or consonants and alternate between them.
Alphabet Position Reference
•
A = 1, B = 2, C = 3, D = 4, E = 5
•
F = 6, G = 7, H = 8, I = 9, J = 10
•
K = 11, L = 12, M = 13, N = 14, O = 15
•
P = 16, Q = 17, R = 18, S = 19, T = 20
•
U = 21, V = 22, W = 23, X = 24, Y = 25, Z = 26
A forward shift moves forward through the alphabet by a fixed number of positions; a backward shift moves in the opposite direction. The shift amount may itself follow a pattern (e.g. increasing by 1 each step).
Constant shift: B, E, H, K, N, … — each letter is 3 positions after the previous one ($+3$ rule).
Increasing shift: A, C, F, J, O, … — the jumps grow by 1 each time ($+2, +3, +4, +5, \ldots$).
Wrap-around: After Z, the alphabet cycles back to A. For example, X, Z, B, D, … — the pattern is $+2$ with wrap-around.
Some letter series use a skip pattern — skipping a fixed number of letters between consecutive terms. A skip of 1 means every other letter; a skip of 2 means every third letter.
Skip-1 pattern: A, C, E, G, I, … — one letter is skipped between each pair.
Skip-2 pattern: A, D, G, J, M, … — two letters are skipped between each pair.
Grouped patterns: Two letters forward, one letter back (e.g. A, B, A, C, A, D, …). The anchor letter A repeats while the second letter advances.
Common Skip Patterns (starting from A)
•
Skip 0 (every letter): A, B, C, D, E, …
•
Skip 1 (every other): A, C, E, G, I, …
•
Skip 2 (every third): A, D, G, J, M, …
•
Skip 3 (every fourth): A, E, I, M, Q, …
•
Skip 4 (every fifth): A, F, K, P, U, …
Symbol Series and Mixed Sequences
A symbol series uses shapes, arrows, or other symbols instead of letters or numbers. The pattern rule may involve rotation, addition of elements, reflection, or changes in size. The strategy is the same: identify what changes from one term to the next.
Rotation: An arrow pointing up → right → down → left follows a 90° clockwise rotation each step.
Element addition/removal: A figure gains one line, dot, or shading each step, or loses one.
Interleaved symbols: Two or more symbols alternate in a cycle — e.g. ★, ●, ★, ●, ★, … is a simple two-symbol cycle.
Size change: Symbols may grow or shrink in a pattern, or change thickness/boldness.
A mixed series combines two or more element types — letters, numbers, and symbols — in a single sequence. Each element type usually follows its own independent pattern within the shared sequence.
Parallel patterns: In A1, C3, E5, G7, … the letters follow $+2$ and the numbers follow $+2$, but independently.
Interleaved sequences: In 2, X, 4, Z, 6, B, 8, D, … odd positions hold an arithmetic number series (2, 4, 6, 8) and even positions hold a letter series (X, Z, B, D).
Combined rules: Sometimes one element encodes information about the other — e.g. the letter's position equals the number.
Strategy for Solving Any Series
1
Step 1: Identify the type — number, letter, symbol, or mixed.
2
Step 2: If mixed, separate each element type into its own sub-sequence.
3
Step 3: Compute differences (arithmetic) or ratios (geometric) for number sub-sequences.
4
Step 4: Convert letters to alphabet positions and repeat Step 3.
5
Step 5: For symbols, list properties (orientation, count, size) and track changes step-by-step.
6
Step 6: Verify your predicted rule on every given term before selecting the answer.
Application Strategy
Worked Examples
Problem Statement
Find the next term in the series: 7, 12, 17, 22, 27, ?
arithmetic sequencecommon difference
Strategic Toolkit
$$a_n = a_1 + (n - 1)d$$
Step 1
Identify the common difference
Subtract each term from the next to check if the difference is constant.
$$d = 12 - 7 = 5, \quad 17 - 12 = 5, \quad 22 - 17 = 5$$
Step 2
Confirm the pattern
The difference is constant at 5, confirming this is an arithmetic sequence with d = 5.
$$d = 5$$
Step 3
Add the common difference to the last term
The 6th term is the 5th term plus the common difference.
$$a_6 = 27 + 5 = 32$$
Problem Statement
Find the 20th term of the arithmetic sequence: 3, 8, 13, 18, …
arithmetic sequencenth term
Strategic Toolkit
$$a_n = a_1 + (n - 1)d$$
Step 1
Identify a₁ and d
The first term is 3. The common difference is 8 minus 3, which equals 5.
$$a_1 = 3, \quad d = 8 - 3 = 5$$
Step 2
Apply the nth term formula
Plug n = 20, a₁ = 3, and d = 5 into the formula.
$$a_{20} = 3 + (20 - 1) \times 5 = 3 + 95 = 98$$
Problem Statement
Find the sum of the first 15 terms of the arithmetic sequence: 4, 9, 14, 19, …
sum of termsarithmetic sequence
Strategic Toolkit
$$S_n = \frac{n}{2}(a_1 + a_n)$$
Step 1
Find a₁, d, and a₁₅
The first term is 4 and the common difference is 5. First compute the 15th term.
$$a_{15} = 4 + (15 - 1) \times 5 = 4 + 70 = 74$$
Step 2
Apply the sum formula
Use the sum formula with n = 15, a₁ = 4, and a₁₅ = 74.
$$S_{15} = \frac{15}{2}(4 + 74) = \frac{15}{2} \times 78 = 15 \times 39 = 585$$
Problem Statement
Find the next term in the series: 3, 6, 12, 24, 48, ?
geometric sequencecommon ratio
Strategic Toolkit
$$a_n = a_1 \cdot r^{\,n-1}$$
Step 1
Identify the common ratio
Divide each term by the previous term to confirm the ratio is constant.
$$r = \frac{6}{3} = 2, \quad \frac{12}{6} = 2, \quad \frac{24}{12} = 2$$
Step 2
Multiply the last term by r
The 6th term is the 5th term multiplied by the common ratio 2.
$$a_6 = 48 \times 2 = 96$$
Problem Statement
Find the 7th term of the geometric sequence: 2, 6, 18, 54, …
geometric sequencecommon ratio
Strategic Toolkit
$$a_n = a_1 \cdot r^{\,n-1}$$
Step 1
Identify a₁ and r
The first term is 2 and the common ratio is 6 divided by 2, which equals 3.
$$a_1 = 2, \quad r = \frac{6}{2} = 3$$
Step 2
Apply the nth term formula
Plug n = 7 into the formula with a₁ = 2 and r = 3.
$$a_7 = 2 \times 3^{6} = 2 \times 729 = 1458$$
Problem Statement
Find the next term in the series: 1, 4, 9, 16, 25, ?
two-step patternsquare numbers
Strategic Toolkit
$$a_n = n^2$$
Step 1
Compute first-level differences
Subtract each term from the next to reveal the pattern in the differences.
$$4 - 1 = 3, \quad 9 - 4 = 5, \quad 16 - 9 = 7, \quad 25 - 16 = 9$$
Step 2
Compute second-level differences
The differences 3, 5, 7, 9 increase by 2 each time — the second-level difference is constant.
$$5 - 3 = 2, \quad 7 - 5 = 2, \quad 9 - 7 = 2$$
Step 3
Predict the next difference and term
The next first-level difference is 9 + 2 = 11. Add it to the last term.
$$a_6 = 25 + 11 = 36 = 6^2$$
Problem Statement
Find the missing term: B, E, H, K, ?, Q
letter seriesalphabet positionforward shift
Strategic Toolkit
Letter position: A = 1, B = 2, …, Z = 26
Step 1
Convert letters to positions
Write the alphabet position for each letter in the series.
$$B = 2, \quad E = 5, \quad H = 8, \quad K = 11, \quad ? = ?, \quad Q = 17$$
Step 2
Identify the pattern
The positions increase by 3 each time: 2 → 5 → 8 → 11. This is an arithmetic sequence with d = 3.
$$d = 5 - 2 = 3, \quad 8 - 5 = 3, \quad 11 - 8 = 3$$
Step 3
Find the missing position
Add 3 to K = 11 to get the missing position.
$$11 + 3 = 14 \rightarrow \text{N}$$
Step 4
Verify
Check that N = 14 followed by Q = 17 also satisfies the +3 rule.
$$17 - 14 = 3 \checkmark$$
Problem Statement
Find the next term in the series: A1, C3, E5, G7, ?
mixed seriesletter seriesarithmetic sequence
Strategic Toolkit
Separate into letter track and number track
Step 1
Separate into two tracks
Extract the letters and numbers into separate sequences.
$$\text{Letters: A, C, E, G} \quad \text{Numbers: 1, 3, 5, 7}$$
Step 2
Find the letter pattern
Convert to positions: A = 1, C = 3, E = 5, G = 7. The positions increase by 2 each time. The next position is 9, which is I.
$$7 + 2 = 9 \rightarrow \text{I}$$
Step 3
Find the number pattern
The numbers increase by 2 each time: 1, 3, 5, 7. The next number is 9.
$$7 + 2 = 9$$
Step 4
Combine the results
The next term pairs the next letter I with the next number 9.
$$\text{I9}$$
Memorize formulas
Arithmetic Sequence — nth Term
Use when the difference between consecutive terms is constant. d is the common difference; n is the position number.
Formula
$$a_n = a_1 + (n - 1)d$$
Arithmetic Series — Sum
Use to find the total of the first n terms. Average the first and last term, then multiply by n.
Formula
$$S_n = \frac{n}{2}(a_1 + a_n)$$
Geometric Sequence — nth Term
Use when the ratio between consecutive terms is constant. r is the common ratio; the first term corresponds to r raised to the 0th power.
Formula
$$a_n = a_1 \cdot r^{\,n-1}$$
Geometric Series — Sum (r ≠ 1)
Use to find the total of the first n terms of a geometric sequence. When 0 < r < 1, rewrite as $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$ for cleaner arithmetic.
Formula
$$S_n = a_1 \cdot \frac{r^{\,n} - 1}{r - 1}$$
Review
1
What is the next term in the series: 3, 8, 13, 18, 23, ?
Ans
28
Review
Tap to reveal
Identify the Pattern
Find the common difference between consecutive terms.
$8 - 3 = 5$, $13 - 8 = 5$, $18 - 13 = 5$, $23 - 18 = 5$
Apply the Rule
Add the common difference to the last term.
$23 + 5 = 28$
2
Find the 15th term of the arithmetic sequence: 5, 9, 13, 17, ...
Ans
61
Review
Tap to reveal
Identify Parameters
Determine the first term and common difference.
$a_1 = 5$, $d = 9 - 5 = 4$
Apply nth Term Formula
Use $a_n = a_1 + (n-1)d$ with $n = 15$.
$a_{15} = 5 + (15 - 1) \times 4 = 5 + 56 = 61$
3
What is the sum of the first 10 terms of the arithmetic series: 2, 7, 12, 17, ...?
Ans
245
Review
Tap to reveal
Find the 10th Term
Using $a_n = a_1 + (n-1)d$ with $a_1 = 2$, $d = 5$.
$a_{10} = 2 + 9 \times 5 = 47$
Calculate the Sum
Apply the arithmetic series sum formula $S_n = \frac{n}{2}(a_1 + a_n)$.
$S_{10} = \frac{10}{2}(2 + 47) = 5 \times 49 = 245$
4
The 5th term of an arithmetic sequence is 23 and the 12th term is 44. What is the common difference?
Ans
3
Review
Tap to reveal
Set Up Equations
Write the nth term formula for both known terms.
$a_1 + 4d = 23$ and $a_1 + 11d = 44$
Subtract and Solve
Subtract the first equation from the second to eliminate $a_1$.
$(a_1 + 11d) - (a_1 + 4d) = 44 - 23 \Rightarrow 7d = 21 \Rightarrow d = 3$
5
What is the next term in the geometric series: 2, 6, 18, 54, ?
Ans
162
Review
Tap to reveal
Find the Common Ratio
Divide consecutive terms to find $r$.
$r = \frac{6}{2} = \frac{18}{6} = \frac{54}{18} = 3$
Calculate Next Term
Multiply the last term by the common ratio.
$54 \times 3 = 162$
6
Find the 6th term of the geometric sequence: 3, 6, 12, 24, ...
Ans
96
Review
Tap to reveal
Identify Parameters
Determine $a_1 = 3$ and $r = 6/3 = 2$.
$a_1 = 3$, $r = 2$
Apply nth Term Formula
Use $a_n = a_1 \cdot r^{n-1}$ for $n = 6$.
$a_6 = 3 \times 2^{5} = 3 \times 32 = 96$
7
What is the sum of the first 5 terms of the geometric series: 1, 3, 9, 27, ...?
Ans
121
Review
Tap to reveal
Identify Parameters
Determine $a_1 = 1$, $r = 3$, $n = 5$.
$a_1 = 1$, $r = 3$
Apply Sum Formula
Use $S_n = a_1 \cdot \frac{r^n - 1}{r - 1}$.
$S_5 = \frac{3^{5} - 1}{3 - 1} = \frac{242}{2} = 121$
8
The 3rd term of a geometric sequence is 20 and the 5th term is 80. What is the common ratio?
Ans
2
Review
Tap to reveal
Set Up Equations
Express both known terms using the geometric nth term formula.
$a_1 r^2 = 20$ and $a_1 r^4 = 80$
Divide and Solve
Divide the second equation by the first to eliminate $a_1$.
$r^2 = \frac{80}{20} = 4 \Rightarrow r = 2$
9
What is the next term in the series: 2, 5, 10, 17, 26, ?
Ans
37
Review
Tap to reveal
Find First Differences
Calculate the difference between consecutive terms.
$5-2=3$, $10-5=5$, $17-10=7$, $26-17=9$
Identify Second Difference
The differences increase by 2 each time — a constant second difference.
Next first difference $= 9 + 2 = 11$
Calculate Next Term
Add the next difference to the last term.
$26 + 11 = 37$
10
What is the next term in the series: 1, 1, 2, 3, 5, 8, ?
Ans
13
Review
Tap to reveal
Identify the Pattern
Each term equals the sum of the two preceding terms.
$1+1=2$, $1+2=3$, $2+3=5$, $3+5=8$
Calculate Next Term
Add the last two terms.
$5 + 8 = 13$
11
What is the next term in the series: 1, 8, 27, 64, ?
Ans
125
Review
Tap to reveal
Identify the Pattern
Recognize each term as a perfect cube.
$1^3 = 1$, $2^3 = 8$, $3^3 = 27$, $4^3 = 64$
Calculate Next Term
Find the cube of the next integer.
$5^3 = 125$
12
What letter comes next in the series: A, D, G, J, ?
Ans
M
Review
Tap to reveal
Convert to Positions
Map each letter to its alphabet position.
A=1, D=4, G=7, J=10
Find the Step Size
Calculate the forward shift between consecutive positions.
$4-1=3$, $7-4=3$, $10-7=3$ → forward shift of 3
Determine Next Letter
Add 3 to the last position and convert back.
$10 + 3 = 13 \rightarrow$ M
13
What letter comes next in the series: Z, W, T, Q, ?
Ans
N
Review
Tap to reveal
Convert to Positions
Map each letter to its alphabet position.
Z=26, W=23, T=20, Q=17
Identify the Direction
Determine the backward shift.
$26-23=3$, $23-20=3$, $20-17=3$ → backward shift of 3
Find the Next Letter
Subtract 3 from the last position.
$17 - 3 = 14 \rightarrow$ N
14
What letter comes next in the series: B, D, G, K, ?
Ans
P
Review
Tap to reveal
Convert to Positions
Map each letter to its position in the alphabet.
B=2, D=4, G=7, K=11
Find the Difference Pattern
The gaps between positions are increasing.
Differences: 2, 3, 4 → next difference = 5
Calculate Next Position
Add the next difference to the last position.
$11 + 5 = 16 \rightarrow$ P
15
What comes next in the series: ACE, BDF, CEG, ?
Ans
DFH
Review
Tap to reveal
Analyze Each Position
Track how each letter position changes across groups.
1st: A→B→C (+1), 2nd: C→D→E (+1), 3rd: E→F→G (+1)
Apply the Shift
Shift each letter of the last group forward by 1.
C→D, E→F, G→H → DFH
16
In the series C, G, K, O, S, ?, what is the position of the missing letter in the alphabet?
Ans
23
Review
Tap to reveal
Convert to Positions
Map each letter to its numerical position.
C=3, G=7, K=11, O=15, S=19
Find the Step Size
The constant difference between consecutive positions.
Step size = 4
Find the Next Position
Add 4 to the last known position.
$19 + 4 = 23$ (letter W)
17
If the pattern ▲, ▼, ●, ▲, ▼, ●, ... continues, what is the 13th symbol?
Ans
▲
Review
Tap to reveal
Identify the Cycle
Determine the repeating unit.
Cycle: ▲, ▼, ● (length = 3)
Find the Position in Cycle
Use modular arithmetic to map the 13th position into the cycle.
$13 \mod 3 = 1 \Rightarrow$ position 1 in the cycle → ▲
18
What comes next in the series: A1, C3, E5, G7, ?
Ans
I9
Review
Tap to reveal
Analyze the Letter Pattern
Track the letter progression separately.
A→C→E→G (+2 each) → next: I
Analyze the Number Pattern
Track the number progression separately.
1→3→5→7 (+2 each) → next: 9
Combine Results
Merge the next letter and number.
I9
19
What comes next in the series: X2, Y4, Z6, A8, ?
Ans
B10
Review
Tap to reveal
Analyze the Letter Pattern
Track the letter progression with alphabet wrapping.
X→Y→Z→A (cyclic +1) → next: B
Analyze the Number Pattern
Track the number progression.
2→4→6→8 (+2 each) → next: 10
Combine Results
Merge the next letter and number.
B10
20
What comes next in the series: 2B, 4D, 8F, 16H, ?
Ans
32J
Review
Tap to reveal
Analyze the Number Pattern
Recognize the geometric progression in numbers.
2, 4, 8, 16 (multiply by 2) → next: $16 \times 2 = 32$
Analyze the Letter Pattern
Track the letter progression.
B→D→F→H (+2 each) → next: J
Combine Results
Merge the next number and letter.
32J
Question 1 of 20
0:00
1
What is the next term in the series: 3, 8, 13, 18, 23, ?