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SSAT Upper Level: Quantitative • Functions and Sequences

Determine Missing Terms in a Sequence

Learn to identify patterns and fill in unknown values in arithmetic, geometric, and other number sequences.

Explore the lesson 14 min read
SSAT Upper Level illustration with a sectioned test book, advanced math, and writing symbols
Inside this lesson 10 sections 5 questions 9 figures

SECTION 1

Historical Context & Motivation

Humans have been fascinated by patterns in numbers for thousands of years. Ancient mathematicians noticed that certain arrangements of objects—pebbles on a counting board, tiles in a mosaic—followed predictable rules. These observations eventually became the formal study of sequences, ordered lists of numbers where each term is connected to the others by a consistent rule. From predicting astronomical events to cracking modern encryption codes, the ability to recognize a pattern and determine a missing piece has been one of the most powerful tools in mathematics.

~300 BCE • Euclid's Elements

Euclid systematically studied geometric progressions and proved that the sum of a geometric series could be calculated with a formula, laying the groundwork for sequence analysis.

~500 CE • Aryabhata's Arithmetic Sequences

Indian mathematician Aryabhata developed formulas for summing arithmetic sequences and applied them to astronomical calculations, including predicting eclipses.

1202 • Fibonacci's Liber Abaci

Leonardo of Pisa (Fibonacci) introduced his famous rabbit-breeding problem, generating the Fibonacci sequence (1, 1, 2, 3, 5, 8, …), where each term is the sum of the two preceding terms.

1785 • Gauss's Summation Insight

As a young student, Carl Friedrich Gauss reportedly summed the integers 1 through 100 in seconds by recognizing the arithmetic pattern, demonstrating the power of sequence formulas.

Today • Standardized Tests & Data Science

Sequence problems appear on exams like the SSAT, SAT, and GRE. In the real world, sequences model population growth, financial interest, computer algorithms, and signal processing.

The core question this lesson addresses is straightforward but essential: given part of a sequence, how do you figure out the rule and use it to find a missing term? Whether the missing value sits at the beginning, middle, or end of the list, the strategy starts with identifying the pattern that connects consecutive terms. Once you have the pattern, finding any missing term becomes a matter of applying it step by step.

SECTION 2

Core Principles & Definitions

Before diving into calculations, you need a solid grasp of the vocabulary and foundational ideas behind sequences. A sequence is an ordered list of numbers, where each number is called a term. The position of each term matters: the first term, second term, third term, and so on are typically written as a₁, a₂, a₃, etc. The key to solving missing-term problems is understanding what type of sequence you are dealing with and what rule governs it.

1

Arithmetic Sequence

Each term is found by adding a constant called the common difference (d) to the previous term. Example: 3, 7, 11, 15, … (d = 4).
2

Geometric Sequence

Each term is found by multiplying by a constant called the common ratio (r). Example: 2, 6, 18, 54, … (r = 3).
3

Recursive / Pattern-Based Sequence

Each term depends on one or more previous terms via a rule, but not necessarily a single add or multiply. Example: 1, 1, 2, 3, 5, 8, … (each term = sum of the two before it).
4

Common Difference vs. Common Ratio

To identify the type, check the differences between consecutive terms first. If they're constant, it's arithmetic. If not, check the ratios. If ratios are constant, it's geometric.
5

Position Notation

The nth term is written aₙ. The first term is a₁, and n represents the position number. This notation lets you write general formulas that work for any position in the sequence.
✦

KEY TAKEAWAY

Think of a sequence like a playlist set to repeat a specific action between songs. An arithmetic sequence is like turning the volume up by the same number of notches each time—steady, predictable steps. A geometric sequence is like doubling the volume each time—the jumps between terms grow rapidly. Once you know which 'control knob' is in play (adding vs. multiplying), finding a missing term is just a matter of turning the knob forward or backward to reach the gap.

SECTION 3

Visual Explanation — Seeing the Pattern

The diagram below shows two sequences plotted on a number line, with a missing term marked by a question mark. The top row shows an arithmetic sequence where terms increase by a constant difference, producing evenly spaced dots. The bottom row shows a geometric sequence where terms increase by a constant ratio, producing dots that spread further apart with each step. Notice how the spacing pattern is the visual clue that reveals which type of sequence you are dealing with.

Arithmetic vs. Geometric SequencesArithmetic (d = 5)3a₁+58a₂+5?a₃+518a₄+523a₅a₃ = 13Geometric (r = 3)2a₁×36a₂×3?a₃×354a₄×3162a₅a₃ = 18Arithmetic: even spacing between dots (constant difference).Geometric: growing spacing between dots (constant ratio).
The arithmetic row (top) uses a common difference of d = 5, so the missing third term is 8 + 5 = 13. The geometric row (bottom) uses a common ratio of r = 3, so the missing third term is 6 × 3 = 18. Notice the dashed gold circle marks the unknown, and the curved arrows show the operation applied between each pair of consecutive terms.

As the diagram makes clear, the first step in any missing-term problem is to look at the gaps between known terms. In the arithmetic example, every gap equals 5. In the geometric example, every term is three times the one before it. Once you confirm the pattern holds across all known terms, you can confidently extend it to fill the blank.

SECTION 4

Mathematical Framework

Translating the visual patterns into formulas gives you a reliable, repeatable method for any sequence problem. Below are the key equations you need.

ARITHMETIC SEQUENCE — NTH TERM

an=a1+(n−1)×daₙ = a₁ + (n − 1) × d

aₙ = the term you want to find, a₁ = the first term, n = position number, d = common difference (the constant added each step). To find d, subtract any term from the term that follows it: d = a₂ − a₁.

GEOMETRIC SEQUENCE — NTH TERM

an=a1×r(n−1)aₙ = a₁ × r⁽ⁿ⁻¹⁾

aₙ = the term you want to find, a₁ = the first term, n = position number, r = common ratio (the constant multiplied each step). To find r, divide any term by the term before it: r = a₂ ÷ a₁.

FINDING d WHEN TERMS ARE NOT CONSECUTIVE

d=(am−ak)÷(m−k)d = (aₘ − aₖ) ÷ (m − k)

If you know two terms that aren't next to each other—say the 2nd and 6th terms—you can still find d by dividing the difference in their values by the number of steps between them.

FINDING r WHEN TERMS ARE NOT CONSECUTIVE

r=(m−k)√(am÷ak)r = ⁽ᵐ⁻ᵏ⁾√(aₘ ÷ aₖ)

If you know the 2nd term is 4 and the 5th term is 108, then r³ = 108 ÷ 4 = 27, so r = ³√27 = 3. You take the (m − k)th root of the ratio of the two known terms.

These formulas handle arithmetic and geometric sequences, which together account for the vast majority of sequence problems on the SSAT. For other sequence types—such as those where each term is the sum of the two before it, or where differences themselves form a sequence—you will rely on pattern recognition rather than a single formula. The strategy is the same: compute the differences (or ratios), look for consistency, and extend the rule to fill the gap.

SECTION 5

Classifying Sequences — A Decision Flowchart

When you see a sequence problem on the SSAT, the first thing to do is classify the sequence. The flowchart below walks you through a systematic approach: start by checking differences, then check ratios, and finally look for more complex patterns like second differences or recursive rules.

Sequence Classification FlowchartGiven: a sequence with a missing termAre the differences between terms constant?YESNOARITHMETIC SEQUENCEUse aₙ = a₁ + (n−1)dAre the ratios between terms constant?YESNOGEOMETRIC SEQUENCEUse aₙ = a₁ × r⁽ⁿ⁻¹⁾Are 2nd differences constant?YESNOQUADRATIC PATTERNDifferences form arithmetic seq.OTHERRecursive, Fibonacci, etc.Tip: On the SSAT, most sequences are arithmetic or geometric.Always check differences first—it's the fastest route to a solution.
Follow the flowchart from top to bottom. Check differences first (arithmetic?), then ratios (geometric?), then second differences (quadratic?). Most SSAT problems resolve in the first two steps.
Sequence TypeExamplePatternHow to Find Missing Term
Arithmetic4, 9, 14, ?, 24d = +5Add 5 to 14 → 19
Geometric3, ?, 48, 192r = ×4Multiply 3 by 4 → 12
Quadratic2, 5, 10, 17, ?Differences: 3, 5, 7, 9Next diff = 9, so 17 + 9 = 26
Fibonacci-type1, 1, 2, 3, 5, ?Each = sum of two prior3 + 5 = 8
Alternating2, −4, 8, −16, ?r = ×(−2)−16 × (−2) = 32

Common sequence types on the SSAT and how to approach each one.

SECTION 6

Worked Example

Let's work through a complete example that mirrors an SSAT problem. We are given the sequence: 5, __, __, 23, 29 and asked to find the two missing terms.

Find the Missing Terms: 5, __, __, 23, 29

  1. 1

    Step 1 — Check Whether It's Arithmetic

    We know two consecutive terms at the end: a₄ = 23 and a₅ = 29. Compute their difference: 29 − 23 = 6. If this is an arithmetic sequence, the common difference d should equal 6 for every pair of consecutive terms. Let's test this hypothesis by checking whether the spacing from a₁ to a₄ is consistent.
    Candidate: d = 6
  2. 2

    Step 2 — Verify with the First and Fourth Terms

    Using the formula d = (aₘ − aₖ) ÷ (m − k), we get d = (23 − 5) ÷ (4 − 1) = 18 ÷ 3 = 6. This matches our candidate, confirming the sequence is arithmetic with d = 6.
    Confirmed: arithmetic sequence with d = 6
  3. 3

    Step 3 — Calculate the Missing Terms

    Now apply d = 6 forward from the first term. a₂ = a₁ + d = 5 + 6 = 11. Then a₃ = a₂ + d = 11 + 6 = 17.
    a₂ = 11, a₃ = 17
  4. 4

    Step 4 — Verify the Complete Sequence

    The complete sequence is 5, 11, 17, 23, 29. Check each consecutive difference: 11 − 5 = 6 ✓, 17 − 11 = 6 ✓, 23 − 17 = 6 ✓, 29 − 23 = 6 ✓. Every difference equals 6, so the answer is confirmed.
    Final answer: 5, 11, 17, 23, 29
💡

Pro Tip for the SSAT

Always verify your answer by plugging the missing terms back in and checking every consecutive pair. This takes only a few seconds and catches arithmetic errors before you commit to an answer choice.

SECTION 7

Strategies, Strengths, and Common Pitfalls

Not every sequence problem on the SSAT is straightforward. Some are designed to trip you up with tricky patterns, negative differences, or ratios less than 1. The table below compares strategies and warns you about the most common mistakes students make.

StrategyWhen to UseCommon Pitfall
Compute differencesFirst approach for any sequence; works if d is constantForgetting to check all pairs—one constant difference isn't enough if there are 4+ known terms
Compute ratiosWhen differences aren't constant; especially useful when terms grow or shrink rapidlyDividing in the wrong order (a₂ ÷ a₁, not a₁ ÷ a₂) or missing a negative ratio
Second differencesWhen first differences form their own arithmetic sequence (quadratic pattern)Stopping after finding first differences and guessing, instead of computing the second layer
Use the nth-term formulaWhen you know the type and need a term far from the given onesOff-by-one errors: the exponent is (n − 1), not n. Position 1 gives exponent 0, not 1.
Back-solve from answer choicesOn multiple choice when direct calculation is slow; plug in each choice and test the patternTesting only one pair instead of verifying the choice works for the entire sequence

Strategies and pitfalls for SSAT sequence problems.

✦

KEY TAKEAWAY

Think of finding a missing term like debugging a text message where some words are replaced by blanks. You read the words around the blank, figure out the sentence's rhythm (the pattern), and the missing word practically fills itself in. The biggest mistake is guessing the rhythm from just one clue—always confirm with multiple pairs before committing.

SECTION 8

Connection to Advanced Concepts

The skills you build with missing-term problems are the foundation for more advanced topics you will encounter in higher math courses. Understanding how arithmetic and geometric sequences behave prepares you for series (the sums of sequences), exponential growth and decay, and even calculus-level ideas about convergence. The table below shows how the concepts in this lesson connect to what comes next.

This LessonAdvanced Extension
Arithmetic sequence (aₙ = a₁ + (n−1)d)Linear functions: y = mx + b, where d is the slope and a₁ is the y-intercept shifted by d
Geometric sequence (aₙ = a₁ × r⁽ⁿ⁻¹⁾)Exponential functions: y = a × bˣ, used to model population growth, compound interest, radioactive decay
Common difference dRate of change (precalculus); first derivative (calculus)
Second differences constantQuadratic functions and parabolas in Algebra 2
Summing terms of a sequenceArithmetic and geometric series formulas; sigma notation

How missing-term skills map to future math topics.

On the SSAT specifically, mastering sequences also helps with function problems and pattern-recognition questions in the quantitative sections. Many word problems about savings accounts, repeated discounts, or distance traveled at a constant speed are secretly arithmetic or geometric sequences in disguise. The pattern-finding muscle you develop here transfers directly to those contexts.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

In the sequence 7, 13, 19, 25, 31, what is the common difference? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8
Show answer
(C) 6. Subtract any consecutive pair: 13 − 7 = 6, 19 − 13 = 6, 25 − 19 = 6, 31 − 25 = 6. The difference is constant at 6. Choice (A) 4 and (B) 5 are simply too small. Choice (D) 7 is the first term, not the difference—a common mix-up. Choice (E) 8 may result from miscounting or adding instead of subtracting.

Question 2: BASIC CALCULATION

What is the missing term in the sequence 4, 12, 36, __, 324? (A) 72 (B) 96 (C) 108 (D) 144 (E) 180
Show answer
(C) 108. Check ratios: 12 ÷ 4 = 3, 36 ÷ 12 = 3. This is a geometric sequence with r = 3. The missing term is 36 × 3 = 108. Verify: 108 × 3 = 324 ✓. Choice (A) 72 results from doubling 36 (using r = 2). Choice (B) 96 might come from adding 60 to 36 without a clear pattern. Choice (D) 144 is 36 × 4, using the wrong ratio. Choice (E) 180 is 36 × 5, also an incorrect ratio.

Question 3: INTERMEDIATE

In an arithmetic sequence, the 3rd term is 14 and the 7th term is 30. What is the 5th term? (A) 18 (B) 20 (C) 22 (D) 24 (E) 26
Show answer
(C) 22. First find d using the formula d = (a₇ − a₃) ÷ (7 − 3) = (30 − 14) ÷ 4 = 16 ÷ 4 = 4. Now count forward from a₃: a₄ = 14 + 4 = 18, a₅ = 18 + 4 = 22. Choice (A) 18 is actually the 4th term—an off-by-one error. Choice (B) 20 might come from averaging 14 and 30 incorrectly. Choice (D) 24 is the 6th term. Choice (E) 26 would require d = 4.8, which doesn't produce integer terms matching the given values.

Question 4: APPLIED

A store reduces the price of a jacket by the same fraction each week. The price starts at $200 in Week 1 and is $50 in Week 3. What is the price in Week 2? (A) $75 (B) $100 (C) $125 (D) $150 (E) $175
Show answer
(B) $100. Since the price is reduced by the same fraction each week, this is a geometric sequence. We have a₁ = 200 and a₃ = 50. Using a₃ = a₁ × r², we get 50 = 200 × r², so r² = 50 ÷ 200 = 0.25, and r = 0.5 (taking the positive root since prices are positive). Therefore a₂ = 200 × 0.5 = 100. Choice (A) $75 might come from incorrectly averaging. Choice (C) $125 is the arithmetic mean of 200 and 50, but this isn't an arithmetic sequence. Choice (D) $150 assumes d = −50 (arithmetic), but that gives a₃ = 100, not 50. Choice (E) $175 is too high and doesn't fit any consistent pattern.

Question 5: CRITICAL THINKING

Consider the sequence 3, 4, 7, 12, 19, __. What is the next term? (A) 24 (B) 26 (C) 28 (D) 30 (E) 33
Show answer
(C) 28. The first differences are 1, 3, 5, 7—these are not constant, so it is not arithmetic. The ratios (4/3, 7/4, 12/7) are not constant either, so it is not geometric. However, the second differences are 2, 2, 2, which are constant. This means the first differences form an arithmetic sequence themselves: the next first difference is 7 + 2 = 9, so the next term is 19 + 9 = 28. Choice (A) 24 would require a next difference of 5, which breaks the pattern. Choice (B) 26 uses a next difference of 7 (repeating instead of continuing). Choice (D) 30 uses a next difference of 11, skipping a step. Choice (E) 33 might come from multiplying 19 by a ratio, but this is not a geometric sequence.

SUMMARY

Lesson Summary

Summary

To determine a missing term in a sequence, start by computing the differences between consecutive known terms. If the differences are constant, you have an arithmetic sequence and can use the formula aₙ = a₁ + (n − 1) × d. If the differences are not constant, check the ratios between consecutive terms. Constant ratios indicate a geometric sequence governed by aₙ = a₁ × r⁽ⁿ⁻¹⁾. For more complex patterns, examine second differences or look for recursive rules like the Fibonacci pattern.

On the SSAT, always verify your answer by plugging the missing term back into the sequence and confirming the pattern holds for every pair. Watch out for off-by-one errors in the nth-term formula, and remember that when two known terms are not consecutive, you can still find d by dividing the value gap by the position gap. With consistent practice, identifying the pattern and solving for a missing term becomes fast and automatic.

Varsity Tutors • ssat-upper-level-quantitative • Determine missing terms in a sequence.