Arithmetic and Geometric Patterns
Help Questions
ISEE Upper Level: Quantitative Reasoning › Arithmetic and Geometric Patterns
A savings plan lists balances: $300, 330, 360, 390, 420$; identify the type and pattern rule.
Arithmetic, with common ratio $r=30$.
Geometric, with common difference $d=30$.
Geometric, with common ratio $r=1.1$.
Arithmetic, with common difference $d=30$.
Explanation
This question tests ISEE Upper Level quantitative reasoning skills, specifically identifying and extending arithmetic or geometric patterns. An arithmetic sequence is defined by a constant difference between terms, whereas a geometric sequence is defined by a constant ratio. Understanding these sequences involves recognizing patterns and applying the correct rules to extend them. In this question, the sequence follows an arithmetic pattern, as indicated by the common difference of 30. Choice B is correct because it accurately follows the pattern rule by identifying it as arithmetic with difference 30. Choice A is incorrect because it represents a common error where students misidentify the sequence type as geometric and confuse the ratio with 1.1. To help students: Encourage practice with identifying sequence types through real-world examples, use visual aids like number lines or graphs to illustrate patterns, and reinforce understanding by having students explain the rule in their own words. Watch for common pitfalls like assuming all sequences are arithmetic or neglecting the context clues provided.
A garden adds 9 plants per row: $14, 23, 32, 41, 50$; calculate the 10th term.
The 10th term is 100, using $r=2$.
The 10th term is 59, reversing $d=9$.
The 10th term is 86, using $d=8$.
The 10th term is 95, using $d=9$.
Explanation
This question tests ISEE Upper Level quantitative reasoning skills, specifically identifying and extending arithmetic or geometric patterns. An arithmetic sequence is defined by a constant difference between terms, whereas a geometric sequence is defined by a constant ratio. Understanding these sequences involves recognizing patterns and applying the correct rules to extend them. In this question, the sequence follows an arithmetic pattern, as indicated by the common difference of 9. Choice A is correct because it accurately follows the pattern rule by calculating the 10th term as 14 + 9*9 = 95. Choice C is incorrect because it represents a common error where students misidentify the sequence type and apply a geometric ratio of 2. To help students: Encourage practice with identifying sequence types through real-world examples, use visual aids like number lines or graphs to illustrate patterns, and reinforce understanding by having students explain the rule in their own words. Watch for common pitfalls like assuming all sequences are arithmetic or neglecting the context clues provided.
An investment adds equal interest: $105, 110, 115, 120, 125$ dollars; what is the next term?
The next term is 120, subtracting $d=5$.
The next term is 250, using $r=2$.
The next term is 128, using $d=3$.
The next term is 130, using $d=5$.
Explanation
This question tests ISEE Upper Level quantitative reasoning skills, specifically identifying and extending arithmetic or geometric patterns. An arithmetic sequence is defined by a constant difference between terms, whereas a geometric sequence is defined by a constant ratio. Understanding these sequences involves recognizing patterns and applying the correct rules to extend them. In this question, the sequence follows an arithmetic pattern, as indicated by the common difference of 5. Choice A is correct because it accurately follows the pattern rule by adding 5 to the last term 125 to get the next term 130. Choice B is incorrect because it represents a common error where students misidentify the sequence type and apply a geometric ratio of 2 instead. To help students: Encourage practice with identifying sequence types through real-world examples, use visual aids like number lines or graphs to illustrate patterns, and reinforce understanding by having students explain the rule in their own words. Watch for common pitfalls like assuming all sequences are arithmetic or neglecting the context clues provided.
A bricklaying plan lists $9, 18, 36, 72, 144$; identify the type and pattern rule.
Arithmetic, with common difference $d=9$.
Arithmetic, with common ratio $r=2$.
Geometric, with common ratio $r=2$.
Geometric, with common difference $d=9$.
Explanation
This question tests ISEE Upper Level quantitative reasoning skills, specifically identifying and extending arithmetic or geometric patterns. An arithmetic sequence is defined by a constant difference between terms, whereas a geometric sequence is defined by a constant ratio. Understanding these sequences involves recognizing patterns and applying the correct rules to extend them. In this question, the sequence follows a geometric pattern, as indicated by the common ratio of 2. Choice B is correct because it accurately follows the pattern rule by identifying it as geometric with ratio 2. Choice A is incorrect because it represents a common error where students misidentify the sequence type as arithmetic and confuse the difference with 9. To help students: Encourage practice with identifying sequence types through real-world examples, use visual aids like number lines or graphs to illustrate patterns, and reinforce understanding by having students explain the rule in their own words. Watch for common pitfalls like assuming all sequences are arithmetic or neglecting the context clues provided.
A wall uses bricks per day: $20, 40, 80, 160, 320$; what is the next term?
The next term is 640, using $r=2$.
The next term is 340, adding 20.
The next term is 160, repeating the fourth term.
The next term is 520, adding 200.
Explanation
This question tests ISEE Upper Level quantitative reasoning skills, specifically identifying and extending arithmetic or geometric patterns. An arithmetic sequence is defined by a constant difference between terms, whereas a geometric sequence is defined by a constant ratio. Understanding these sequences involves recognizing patterns and applying the correct rules to extend them. In this question, the sequence follows a geometric pattern, as indicated by the common ratio of 2. Choice B is correct because it accurately follows the pattern rule by multiplying the last term 320 by 2 to get the next term 640. Choice A is incorrect because it represents a common error where students misidentify the sequence type and add a varying difference like 20 instead of multiplying. To help students: Encourage practice with identifying sequence types through real-world examples, use visual aids like number lines or graphs to illustrate patterns, and reinforce understanding by having students explain the rule in their own words. Watch for common pitfalls like assuming all sequences are arithmetic or neglecting the context clues provided.
A construction crew lays bricks: $6, 12, 24, 48, 96$; which option completes the sequence correctly?
Next is 90, subtracting 6 each day.
Next is 102, adding 6 each day.
Next is 144, multiplying by 1.5.
Next is 192, multiplying by 2.
Explanation
This question tests ISEE Upper Level quantitative reasoning skills, specifically identifying and extending arithmetic or geometric patterns. An arithmetic sequence is defined by a constant difference between terms, whereas a geometric sequence is defined by a constant ratio. Understanding these sequences involves recognizing patterns and applying the correct rules to extend them. In this question, the sequence follows a geometric pattern, as indicated by the common ratio of 2. Choice B is correct because it accurately follows the pattern rule by multiplying the last term 96 by 2 to get the next term 192. Choice A is incorrect because it represents a common error where students misidentify the sequence type and add a difference like 6 instead. To help students: Encourage practice with identifying sequence types through real-world examples, use visual aids like number lines or graphs to illustrate patterns, and reinforce understanding by having students explain the rule in their own words. Watch for common pitfalls like assuming all sequences are arithmetic or neglecting the context clues provided.
An investment grows by equal deposits: $50, 58, 66, 74, 82$; what is the next term?
The next term is 88, using $d=6$.
The next term is 164, using $r=2$.
The next term is 74, subtracting $d=8$.
The next term is 90, using $d=8$.
Explanation
This question tests ISEE Upper Level quantitative reasoning skills, specifically identifying and extending arithmetic or geometric patterns. An arithmetic sequence is defined by a constant difference between terms, whereas a geometric sequence is defined by a constant ratio. Understanding these sequences involves recognizing patterns and applying the correct rules to extend them. In this question, the sequence follows an arithmetic pattern, as indicated by the common difference of 8. Choice A is correct because it accurately follows the pattern rule by adding 8 to the last term 82 to get the next term 90. Choice B is incorrect because it represents a common error where students misidentify the sequence type and apply a geometric ratio of 2. To help students: Encourage practice with identifying sequence types through real-world examples, use visual aids like number lines or graphs to illustrate patterns, and reinforce understanding by having students explain the rule in their own words. Watch for common pitfalls like assuming all sequences are arithmetic or neglecting the context clues provided.
An investment ledger shows $80, 92, 104, 116, 128$ dollars; which option completes the sequence correctly?
Next is 116, subtracting 12.
Next is 256, multiplying by 2.
Next is 140, adding 12.
Next is 138, adding 10.
Explanation
This question tests ISEE Upper Level quantitative reasoning skills, specifically identifying and extending arithmetic or geometric patterns. An arithmetic sequence is defined by a constant difference between terms, whereas a geometric sequence is defined by a constant ratio. Understanding these sequences involves recognizing patterns and applying the correct rules to extend them. In this question, the sequence follows an arithmetic pattern, as indicated by the common difference of 12. Choice A is correct because it accurately follows the pattern rule by adding 12 to the last term 128 to get the next term 140. Choice B is incorrect because it represents a common error where students misidentify the sequence type and multiply by 2. To help students: Encourage practice with identifying sequence types through real-world examples, use visual aids like number lines or graphs to illustrate patterns, and reinforce understanding by having students explain the rule in their own words. Watch for common pitfalls like assuming all sequences are arithmetic or neglecting the context clues provided.
A savings account balance increases: $200, 215, 230, 245, 260$; what is the next term?
The next term is 520, using $r=2$.
The next term is 275, using $d=15$.
The next term is 270, using $d=10$.
The next term is 245, reversing $d=15$.
Explanation
This question tests ISEE Upper Level quantitative reasoning skills, specifically identifying and extending arithmetic or geometric patterns. An arithmetic sequence is defined by a constant difference between terms, whereas a geometric sequence is defined by a constant ratio. Understanding these sequences involves recognizing patterns and applying the correct rules to extend them. In this question, the sequence follows an arithmetic pattern, as indicated by the common difference of 15. Choice A is correct because it accurately follows the pattern rule by adding 15 to the last term 260 to get the next term 275. Choice B is incorrect because it represents a common error where students misidentify the sequence type and apply a geometric ratio of 2. To help students: Encourage practice with identifying sequence types through real-world examples, use visual aids like number lines or graphs to illustrate patterns, and reinforce understanding by having students explain the rule in their own words. Watch for common pitfalls like assuming all sequences are arithmetic or neglecting the context clues provided.
In a garden, plants per row are $7, 11, 15, 19, 23$; calculate the 10th term.
The 10th term is 39, subtracting $d=4$.
The 10th term is 56, using $r=2$.
The 10th term is 43, using $d=4$.
The 10th term is 47, using $d=5$.
Explanation
This question tests ISEE Upper Level quantitative reasoning skills, specifically identifying and extending arithmetic or geometric patterns. An arithmetic sequence is defined by a constant difference between terms, whereas a geometric sequence is defined by a constant ratio. Understanding these sequences involves recognizing patterns and applying the correct rules to extend them. In this question, the sequence follows an arithmetic pattern, as indicated by the common difference of 4. Choice A is correct because it accurately follows the pattern rule by calculating the 10th term as 7 + 9*4 = 43. Choice C is incorrect because it represents a common error where students misidentify the sequence type and apply a geometric ratio of 2. To help students: Encourage practice with identifying sequence types through real-world examples, use visual aids like number lines or graphs to illustrate patterns, and reinforce understanding by having students explain the rule in their own words. Watch for common pitfalls like assuming all sequences are arithmetic or neglecting the context clues provided.