Recognize a pattern in a sequence of numbers - CogAT Grades 3 Plus

1. Based on the graph shown, what is the value of the 5th term in the sequence?
   Answer: 16 The y-values rise by 3 each step (4, 7, 10, 13). Adding 3 to 13 yields 16. The other options add 2, 4, or 5, which would break the pattern.
2. In a number sequence, the 3rd term is 17, the 5th term is 37, and the 7th term is 57. If this pattern continues, what is the 12th term?
   Answer: 107 From 3rd to 5th term: 37-17=20 over 2 positions, so +10 per position. From 5th to 7th: 57-37=20 over 2 positions, confirming +10 per position. Working backwards from 3rd term (17): 1st term = 17-20 = -3. The sequence is -3, 7, 17, 27, 37, 47, 57, 67, 77, 87, 97, 107. So 12th term is 107. Choice B (117) adds one extra +10. Choice C (97) is the 11th term. Choice D (127) adds two extra +10s.
3. A sequence has the pattern: 6, 18, 54, 162, 486, ... If you were to write the 8th term of this sequence, how many digits would it contain?
   Answer: 5 digits When you encounter a sequence problem, your first step is identifying the pattern. Looking at this sequence: 6, 18, 54, 162, 486..., you can find the relationship between consecutive terms by dividing each term by the previous one: $18 ÷ 6 = 3$, $54 ÷ 18 = 3$, $162 ÷ 54 = 3$. This is a geometric sequence where each term is multiplied by 3. To find the 8th term, you need to continue the pattern. The 6th term is $486 × 3 = 1,458$, and the 7th term is $1,458 × 3 = 4,374$. The 8th term is $4,374 × 3 = 13,122$. Counting the digits in 13,122 gives you 5 digits. Looking at the wrong answers: Choice A suggests 4 digits, which would be a number between 1,000 and 9,999 - this falls short of our calculated value. Choice B claims 3 digits (100-999), which is far too small given how rapidly this sequence grows. Choice C suggests 6 digits, which would mean a number of at least 100,000 - this overshoots our actual 8th term. The correct answer is D) 5 digits, since 13,122 contains exactly five digits. For CogAT sequence problems, always identify the pattern first (addition, multiplication, or more complex relationships), then systematically work through the terms. Don't try to jump ahead too quickly - geometric sequences with multiplication factors greater than 2 grow very rapidly, and it's easy to lose track if you skip steps.
4. A sequence starts: 1, 4, 10, 22, 46, ... Each term after the first is created using the same rule. What is the 7th term in this sequence?
   Answer: 190 Looking at the pattern: 1→4 (×4+0), but let's check systematically. 1→4: ×4+0? No. Let's try: 1×2+2=4, 4×2+2=10, 10×2+2=22, 22×2+2=46. This works! Each term is doubled and then 2 is added. Continuing: 46×2+2=94 (6th term), 94×2+2=190 (7th term). Choice B (184) might come from 94×2-4. Choice C (196) might come from 94×2+8. Choice D (188) might come from 94×2-0.
5. Refer to the stair diagram. Each step shows a number. What number should go on the missing top step marked with a question mark (?)?
   Answer: 60 The numbers decrease by 10 as the steps go up (100, 90, 80, 70). Subtracting 10 from 70 gives 60. 50, 65, and 75 do not keep the −10 pattern.
6. Consider the sequence: 100, 94, 85, 73, 58, 40, ... What is the next term that should follow 40?
   Answer: 19 Looking at the differences between consecutive terms: 100→94 (-6), 94→85 (-9), 85→73 (-12), 73→58 (-15), 58→40 (-18). The differences form the pattern -6, -9, -12, -15, -18, which decreases by 3 each time. So the next difference should be -21, making the next term 40-21=19. Choice A (22) results from using -18 again. Choice C (25) might come from adding instead of subtracting. Choice D (16) results from using a difference of -24.
7. Look at this sequence: 2, 6, 18, 54, 162, ... Two students identified different patterns. Maria says each term is multiplied by 3. Josh says each term is the previous term plus 4, then that result is multiplied by some number. Who identified the correct pattern?
   Answer: Maria is correct; the pattern is multiply by 3 When you encounter number sequences on the CogAT, you need to identify the consistent pattern that generates each term from the previous one. The key is testing each proposed rule systematically across all terms. Let's examine Maria's pattern first: multiply each term by 3. Starting with 2: $2 \times 3 = 6$, then $6 \times 3 = 18$, then $18 \times 3 = 54$, and $54 \times 3 = 162$. This works perfectly for every transition in the sequence. Now let's test Josh's pattern: add 4, then multiply by some number. From 2 to 6: $2 + 4 = 6$, then multiply by 1 to get 6. From 6 to 18: $6 + 4 = 10$, then multiply by 1.8 to get 18. From 18 to 54: $18 + 4 = 22$, then multiply by approximately 2.45 to get 54. Josh's pattern requires different multipliers each time, which means it's not actually a consistent pattern. Answer D is correct because Maria identified the true pattern: multiply by 3. Answer A is wrong because there clearly is a working rule. Answer B is incorrect because Josh's approach requires constantly changing multipliers, making it unreliable and not a true pattern. Answer C is wrong because while Maria's pattern works consistently, Josh's doesn't represent a single, stable rule. Remember: effective sequence patterns use the same operation with the same numbers throughout. If your proposed rule requires changing values at each step, look for a simpler, more consistent pattern.
8. Use the figure. The rectangles show a sequence from left to right. What number should replace the question mark (?) in the last rectangle?
   Answer: 2 Each rectangle's number is half of the previous one (32→16→8→4). Halving 4 gives 2. 3, 4, and 5 do not match the ÷2 rule.
9. A sequence follows this pattern: 2, 8, 32, 128, 512, ... If this pattern continues, which number would appear exactly three positions after 512 in the sequence?
   Answer: 32,768 The pattern shows each term is multiplied by 4: 2×4=8, 8×4=32, 32×4=128, 128×4=512. Continuing three positions after 512: 512×4=2,048 (1st after), 2,048×4=8,192 (2nd after), 8,192×4=32,768 (3rd after).
10. The sequence 3, 7, 15, 31, 63, ... follows a specific pattern. Based on this pattern, what number comes immediately before 3 in this sequence?
    Answer: 1 The pattern is: each term equals 2×(previous term)+1. Check: 3×2+1=7, 7×2+1=15, 15×2+1=31, 31×2+1=63. To find what comes before 3, we solve: 2x+1=3, so 2x=2, so x=1. Choice A (0) would give 2×0+1=1≠3. Choice C (-1) would give 2×(-1)+1=-1≠3. Choice D (2) would give 2×2+1=5≠3.

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