Free Practice Question
PS · Problem Solving
705-805
Sequences
An antique store has a collection of eight clocks. At a particular moment, the displayed times on seven of the eight clocks were as follows: 1:55 pm, 2:03 pm, 2:11 pm, 2:24 pm, 2:45 pm, 3:19 pm and 4:14 pm. If the displayed times of all eight clocks form an evenly spaced sequence (arithmetic progression), then what was the displayed time on the remaining clock?
Answer Choices
Correct answer marked belowA
1:53 pm
1:58 pm
Correct
C
2:18 pm
D
3:08 pm
E
5:08 pm
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Official Explanation
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The displayed times of all eight clocks form an arithmetic progression.
Given seven times: 1:55 pm, 2:03 pm, 2:11 pm, 2:24 pm, 2:45 pm, 3:19 pm, 4:14 pm.
Convert all times to minutes after 1:00 pm for easier calculation:
- 1:55 pm = 55 minutes
- 2:03 pm = 63 minutes
- 2:11 pm = 71 minutes
- 2:24 pm = 84 minutes
- 2:45 pm = 105 minutes
- 3:19 pm = 139 minutes
- 4:14 pm = 194 minutes
Now examine the differences between consecutive given times:
63 − 55 = 8
71 − 63 = 8
84 − 71 = 13
105 − 84 = 21
139 − 105 = 34
194 − 139 = 55
The differences are: 8, 8, 13, 21, 34, 55.
Notice these are Fibonacci numbers: 8, 8, 13, 21, 34, 55.
In a Fibonacci sequence, each term is the sum of the two preceding terms.
Here, 8 + 8 = 16, but the next difference is 13 — not matching.
However, if we look at the pattern, the differences themselves seem to follow a Fibonacci-like pattern after the first two 8's: 8, 8, 13, 21, 34, 55.
For an arithmetic progression, the common difference should be constant.
But here the differences are not constant, meaning the given seven times are not consecutive terms of the AP — one term is missing.
We need to find where the missing clock fits so that all eight terms are equally spaced.
Let the eight times in AP be: .
The given seven times are some seven of these eight.
List the minute values in ascending order: 55, 63, 71, 84, 105, 139, 194.
The total range is 194 − 55 = 139 minutes.
In an eight-term AP, the total span from first to last is .
So ⇒ , not an integer — suspicious, but d need not be integer in minutes.
Instead, notice the differences between the given times: 8, 8, 13, 21, 34, 55.
These look like Fibonacci numbers.
If the sequence of eight times is an AP, then the differences between consecutive terms in sorted order should all equal d.
But here the differences vary, so one difference is split into two parts because the missing term lies between two given terms.
Test option B: 1:58 pm = 58 minutes.
Insert 58 into the sorted list: 55, 58, 63, 71, 84, 105, 139, 194.
Now compute differences:
58 − 55 = 3
63 − 58 = 5
71 − 63 = 8
84 − 71 = 13
105 − 84 = 21
139 − 105 = 34
194 − 139 = 55
The differences are: 3, 5, 8, 13, 21, 34, 55.
These are consecutive Fibonacci numbers (starting from 3).
In an arithmetic progression, differences should be constant, but here they are increasing — so this is not an AP.
Wait — the problem says "evenly spaced sequence (arithmetic progression)", meaning constant difference.
But the given differences with 58 inserted are not constant.…
Given seven times: 1:55 pm, 2:03 pm, 2:11 pm, 2:24 pm, 2:45 pm, 3:19 pm, 4:14 pm.
Convert all times to minutes after 1:00 pm for easier calculation:
- 1:55 pm = 55 minutes
- 2:03 pm = 63 minutes
- 2:11 pm = 71 minutes
- 2:24 pm = 84 minutes
- 2:45 pm = 105 minutes
- 3:19 pm = 139 minutes
- 4:14 pm = 194 minutes
Now examine the differences between consecutive given times:
63 − 55 = 8
71 − 63 = 8
84 − 71 = 13
105 − 84 = 21
139 − 105 = 34
194 − 139 = 55
The differences are: 8, 8, 13, 21, 34, 55.
Notice these are Fibonacci numbers: 8, 8, 13, 21, 34, 55.
In a Fibonacci sequence, each term is the sum of the two preceding terms.
Here, 8 + 8 = 16, but the next difference is 13 — not matching.
However, if we look at the pattern, the differences themselves seem to follow a Fibonacci-like pattern after the first two 8's: 8, 8, 13, 21, 34, 55.
For an arithmetic progression, the common difference should be constant.
But here the differences are not constant, meaning the given seven times are not consecutive terms of the AP — one term is missing.
We need to find where the missing clock fits so that all eight terms are equally spaced.
Let the eight times in AP be: .
The given seven times are some seven of these eight.
List the minute values in ascending order: 55, 63, 71, 84, 105, 139, 194.
The total range is 194 − 55 = 139 minutes.
In an eight-term AP, the total span from first to last is .
So ⇒ , not an integer — suspicious, but d need not be integer in minutes.
Instead, notice the differences between the given times: 8, 8, 13, 21, 34, 55.
These look like Fibonacci numbers.
If the sequence of eight times is an AP, then the differences between consecutive terms in sorted order should all equal d.
But here the differences vary, so one difference is split into two parts because the missing term lies between two given terms.
Test option B: 1:58 pm = 58 minutes.
Insert 58 into the sorted list: 55, 58, 63, 71, 84, 105, 139, 194.
Now compute differences:
58 − 55 = 3
63 − 58 = 5
71 − 63 = 8
84 − 71 = 13
105 − 84 = 21
139 − 105 = 34
194 − 139 = 55
The differences are: 3, 5, 8, 13, 21, 34, 55.
These are consecutive Fibonacci numbers (starting from 3).
In an arithmetic progression, differences should be constant, but here they are increasing — so this is not an AP.
Wait — the problem says "evenly spaced sequence (arithmetic progression)", meaning constant difference.
But the given differences with 58 inserted are not constant.…
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Sign in to commentWhat this question tests
What it tests
Your ability to spot and extend arithmetic and geometric sequences and work with recursive definitions.
Common trap
Off-by-one errors in indexing (a₀ vs a₁) or confusing an arithmetic difference with a geometric ratio.
Details
Difficulty
705-805
Type
PS
Category