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655-705 Level|   Math Related|         
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MKeerthu
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 15:12
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The answers are 8 and 13.

When S4 = 5, which is not divisible by 4. So, S5 = 5+3 = 8. Now 8 is divisible by 4. So S6 = 8+2=10. 10 is not present in the available options

When S4= 8 which is div by 4, So S5 = 8+2 =10, now 10 is not div by 4. So 10 +3 = 13.

We have 8 and 13 in the options. Try for other numbers too, you will not find a pair.
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 15:54
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A sequence of integers is defined using the following logic: If \(s_n\) is divisible by 4, then \(s_{n+1} = s_n + 2\). If \(s_n\) is not divisible by 4, then \(s_{n+1} = s_n + 3\).

Select values for \(s_4\) and \(s_6 \)that are jointly consistent with these conditions. Make only two selections, one in each column.
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S4 = 8, S6 = 13

Let us say S4 = 8

8/4=2 (divisible)

So, S5 = S4 + 2 = 8 + 2 = 10

10/4=2.5 (not divisible)

So, S6 = S5 + 3 = 10 + 3 = 13

These values satisfy the conditions.
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 16:01
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Let's start by considering the possible values for s4 and how they lead to the value of s6.

First, we try s4=5. Since 5 is not divisible by 4, we would add 3 to get s5=5+3=8. Now, 8 is divisible by 4, so according to the rules, we would add 2 to get s6=8+2=10. However, 10 is not one of the options for s6, so s4=5 is not a valid choice.

Next, we try s4=8. Since 8 is divisible by 4, we would add 2 to get s5=8+2=10. Since 10 is not divisible by 4, we add 3 to get s6=10+3=13, which is a valid option for s6s6.

Therefore, the values s4=8 and s6=13 are the correct answers.

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A sequence of integers is defined using the following logic: If \(s_n\) is divisible by 4, then \(s_{n+1} = s_n + 2\). If \(s_n\) is not divisible by 4, then \(s_{n+1} = s_n + 3\).

Select values for \(s_4\) and \(s_6 \)that are jointly consistent with these conditions. Make only two selections, one in each column.
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 16:30
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Here I'd first go to the statements and go trial and error.

So for the values were S4 is not divisible by 4 -> S4 = 5 & S5 = s4 + 3 = 5 + 3 = 8 and finally making it S6 = 8 + 3 = 11, which is the couple of solutions.
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 16:58
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The only values that match the rules are:
s4=8
s5=10
s6=13
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 17:24
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A sequence of integers is defined using the following logic: If \(s_n\) is divisible by 4, then \(s_{n+1} = s_n + 2\). If \(s_n\) is not divisible by 4, then \(s_{n+1} = s_n + 3\).

Select values for \(s_4\) and \(s_6 \)that are jointly consistent with these conditions. Make only two selections, one in each column.
Show more

Easiest way in this one is to start plugging in values. The 4 and 6 are irrelevant, besides the distance between them. It could just as easily be the first and third integers in the sequence.

Ultimately, plugging 8 for S4 means that S5 will be 10 (8+2, since 8 was divisible by 4). S5 is now 10 and isn't divisible by 4, so we add 3 to get 13 for S6. Those are both valid options, so choose S4=8 and S6=13
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 18:32
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If s4=8 then s5 is 8+2=10
If s5=10 then s6 is 10+3=13

Answers 8 and 13
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 18:41
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\(s_4\) = 8 and \(s_6\) = 13.

We can work by elimination fairly quickly:
If \(s_4\) = 5, then \(s_5\) = 8, and \(s_6 \) = 10, which doesn't work based on the numbers we're given.
If \(s_4\) = 8, then \(s_5\) = 10, and \(s_6 \) = 13, which does work based on the numbers we're given.
If \(s_4\) = 13, then \(s_5\) = 16, and \(s_6 \) = 18, which doesn't work based on the numbers we're given.
If \(s_4\) = 16, then \(s_5\) = 18, and \(s_6 \) = 21, which doesn't work based on the numbers we're given.
If \(s_4\) = 22, then \(s_5\) = 25, and \(s_6 \) = 28, which doesn't work based on the numbers we're given.

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A sequence of integers is defined using the following logic: If \(s_n\) is divisible by 4, then \(s_{n+1} = s_n + 2\). If \(s_n\) is not divisible by 4, then \(s_{n+1} = s_n + 3\).

Select values for \(s_4\) and \(s_6 \)that are jointly consistent with these conditions. Make only two selections, one in each column.
Show more
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 19:54
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A sequence of integers is defined using the following logic: If \(s_n\) is divisible by 4, then \(s_{n+1} = s_n + 2\). If \(s_n\) is not divisible by 4, then \(s_{n+1} = s_n + 3\).

Select values for \(s_4\) and \(s_6 \)that are jointly consistent with these conditions. Make only two selections, one in each column.
Show more
We start looking at the options,
If we start with s4 as 5,
s5 = 5 + 3 = 8 (since 5 is not divisible by 4)
s6 = 8 + 2 = 10. (since 8 is divisible by 4)
Not possible

If we start with s4 as 8,
s5 = 8 + 2 = 10 (since 8 is divisible by 4)
s6 = 10 + 3 = 13. (since 10 is not divisible by 4)
Works.
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 20:41
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A sequence of integers is defined using the following logic: If \(s_n\) is divisible by 4, then \(s_{n+1} = s_n + 2\). If \(s_n\) is not divisible by 4, then \(s_{n+1} = s_n + 3\).

Select values for \(s_4\) and \(s_6 \)that are jointly consistent with these conditions. Make only two selections, one in each column.
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Brute force the solution by trying all possibilities for s_4
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 20:50
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A sequence of integers is defined using the following logic: If \(s_n\) is divisible by 4, then \(s_{n+1} = s_n + 2\). If \(s_n\) is not divisible by 4, then \(s_{n+1} = s_n + 3\).

Select values for \(s_4\) and \(s_6 \)that are jointly consistent with these conditions. Make only two selections, one in each column.
Show more
If \(s_n\) is divisible by 4
[ltr]

\(s_6 \) = \(s_4\) + 4

If \(s_n\) is not divisible by 4

\(s_6 \) = \(s_4\) + 6
[/ltr]
Possible combination

(1) 8 and 12
(2) 12 and 16
(3) 16 and 22

(1) If S4 = 8, push back to S1, S1 + 8 = S4, S1 = 0 => not divisible by 4 (x)
(2) If S4 = 12, push back to S1, S1 + 8 = S4, S1 = 4 => divisible by 4 (✔)
(3) S6 - S4 = 6, but S4 is divisible by 4, so it is contradict to the question (x)

Therefore S4 = 12, S6 = 16
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 21:07
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Step-by-Step Reasoning:

  1. The sequence is defined by:
    • If Sn is divisible by 4, then Sn + 1 = Sn + 2s.
    • If Sn is not divisible by 4, then Sn + 1 = Sn + 3s.
    We are given a set of possible values for S4 and S6 and must pick one value for S4 and one for S6 from the provided options that could occur in the same sequence.
  2. Test each possible S4 from the given options and see if we can arrive at an S6 also from the options:
    Try S4 = 8:
    • 8 is divisible by 4, so S5 = 8 + 2 = 10.
    • 10 is not divisible by 4, so S6 = 10 + 3 = 13.
    With S4 = 8, we get S6 = 13. Both 8 and 13 are in the provided lists of possible values.
    We’ve found a consistent pair: S4 = 8 and S6 = 13.
  3. Since we’ve found a working pair, we do not need to test the others further for this problem.
Answer:
  • S4 = 8
  • S6 = 13
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 21:18
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(1) if s4 is divisible by 4 then s5= s4+2 implies s5 will not be divisible by 4 which implies s6=s5+3 therefore s6 = s4+5

(2) s4 is not divisible by 4 then s5=s4+3, here we have 2 cases s5 is divisible by 4 or not therefore s6 = s4+5 or s6 = s4+6

so for any value of s4 based on its divisibility of 4 check if a corresponding value of s6 exists. answer is 8 and 13
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 21:36
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s4= 8
s6= 12

If sn is divisible by 4, then sn+1 = sn + 2. ----> this works here
If sn is not divisible by 4, then sn+1 = sn + 3.
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 22:04
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Easiest way for me is just to start testing out the number
From the formulas, I noticed that the sequence is increasing. Therefore, I start testing S4 from the smallest value
- If S4 = 5 -> S5 = 8 -> S6 = 11 (not in answer choice -> remove)
- S4 = 8 -> S5 = 10 -> S6 = 13. Both S4 and S6 are present - choose these options
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 22:10
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If divisible by 4: S(n+1) = S(n) + 2
If not divisible by 4: S(n+1) = S(n) + 3

Putting values for S4 from the options we get valid pair as (8, 13).
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 22:10
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S4-E,
S5-F
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 22:26
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Let's take a look at an example:

S1 = 4 --> S2 = 6 --> S3 = 9 --> S4 = 12 --> S5 = 14 -> S6 = 17 --> S7 = 20

The pattern above shows that:

S4 & S6 must be either 5 or 6 different.
IF S6 - S4 = 5, either term must be divisible by 4
IF S6 - S4 = 6, the larger term S6 must be divisible by 4, while the smaller term S4 cannot be divisible by 4.

Answer: S4 = 8, S6 = 13
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 22:48
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A sequence of integers is defined using the following logic: If \(s_n\) is divisible by 4, then \(s_{n+1} = s_n + 2\). If \(s_n\) is not divisible by 4, then \(s_{n+1} = s_n + 3\).

Select values for \(s_4\) and \(s_6 \)that are jointly consistent with these conditions. Make only two selections, one in each column.
Show more

\(s_4 = 5\\
s_5 = 5 + 3 = 8\\
s_6 = 8 + 2 = 10\)

No combination exists

\(s_4 = 8\\
s_5 = 8 + 2 = 10 \\
s_6 = 10 + 3 = 13\)

Hence, column A = 8 & column B = 13
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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers 16 Dec 2024, 23:25
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In \(s_4\) column, numbers which are divisible by 4 are 8, 12 & 16.

Case 1: \(s_4\) = 8
As \(s_n\) is divisible by 4, we will use \(s_n_+_1_\) = \(s_n\) + 2
\(s_n_+_1_\) = \(s_5\) = \(s_4\) + 2 = 8 + 2 = 10, which is not divisible for 4.

So now we will use \(s_n_+_1_\) = \(s_n\) + 3

We got \(s_5\) = 10. Substituting n = 5 in above equation,
\(s_5_+_1_\) = \(s_6\) = \(s_5\) + 3 = 10 + 3 = 13

\(s_4\) = 8
\(s_6\) = 13
This matches the available options.

Case 2: \(s_4\) = 12
Using the similar steps done for case 1, we obtain -
\(s_4\) = 12
\(s_6\) = 17

Case 3: \(s_4\) = 16
Using the similar steps done for case 1, we obtain -
\(s_4\) = 16
\(s_6\) = 21
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