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12 Days of Christmas GMAT Competition - Day 1: A sequence of integers

1 2 3 4 5

Lizaza

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16 Dec 2024, 23:33

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Here, the important thing is that we establish how the numbers will be changing.
Let's visualise this, starting with 1:

1 [not divisible] => $1+3=4$
4 [divisible] => $4+2=6$
6 [not divisible] => $6+3=9$
9 [not divisible] => $9+3=12$
12 [divisible] => $12+2=14 $, etc.

Therefore, if the starting number $s_4 $ is divisible, then $s_6$ will be equal to $s_4 + 5$
And if it's not divisible, then there're two options possible: $s_6=s_4+6 $ and $s_4 + 5$ (as we see from the above variants, where $6=1+5$, but $12=6+6$).

So, checking the answers suggested:

$s_4=5$ (indivisible), so $s_6=s_4+5=10 $ (not suitable) and $s_6=s_4+6=11 $ (also not suitable)
$s_4=8$ (divisible), so $s_6=s_4+5=13 $ (actually suitable)

Hence, the answers are: $s_4=8, s_6=13$

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17 Dec 2024, 00:09

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Bunuel

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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.

Let's assume from that number is divisible by 4
Available options are 8, 12 & 16
for 8 S5th term will be 10 & S6th will be 13

both of these terms satisfy the eqns & available in option

IMO 8 & 13

SafSin28

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Updated on: 17 Dec 2024, 22:24

Originally posted by SafSin28 on 17 Dec 2024, 00:28.
Last edited by SafSin28 on 17 Dec 2024, 22:24, edited 1 time in total.

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Good question

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17 Dec 2024, 00:38

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when $s_4=8$, $s_4$ is divisible by $4$, hence $s_5=s_4+2=10$

$s_5=10$ and it is not divisible by $4$, hence $s_6=s_5+3=13$

Answer:

$s_4=8$

$s_6=13$

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17 Dec 2024, 00:56

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Let's solve from the options.
If s4=5, s5=8, s6=10,
If s4=8, s5=10, s6=13,
If s4=12, s5=14, s6=17,
If s4=13, s5=16, s6=18,
Is s4=16, s5=18, s6=21,
If s4=22, s5=25, s6=28.

Clearly only one case is jointly consistent with the given options, Therefore s4=8, s6=13.

Bunuel

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17 Dec 2024, 01:16

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Hi All,

According to me,

If we try by hit and trail with the options for s4 , the sequence fits correct when s4=8 , as 8 is divisible by 4 so s5=8+2-->10 and 10 is not divisible by 4 so s6=10+3 --> 13

therefore s4=8 and s6=13

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17 Dec 2024, 01:40

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Bunuel

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Trial and Error method will work here

lets first check value of S4 by considering each values
1st - S4 = 5 (not divisible by 4) Therefore S5 = S4 + 3 = 5+3 = 8 (divisible by 4) Therefore S6 = S5 + 2 = 8+2 = 10 (Not there in table So S4=5 not possible solution)
Similarly,
2nd - S4 = 8 (divisible by 4), We will get S5 = S4+2 = 10 (not divisible by 4) S6 = S5+3 = 10+3 = 13 (There in the table) lets keep this solution and check the rest.
3rd - S4 = 12, S5 = S4 + 2 = 14 Now S6 = S5 + 3 = 17 (Not there in table)
4th - S4 = 13, S5 = S4 + 3 = 16 Now S6 = S5 + 2 = 18 (Not there in table)
5th - S4 = 16, S5 = S4 + 2 = 18 Now S6 = S5 + 3 = 21 (Not there in table)
Plus there is no point in checking S4 = 22 as S6 will be higher so the second condition satisfies.
Therefore, S4 = 8 and S6 = 13.

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17 Dec 2024, 02:47

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start with s3 as 1 and work till s4 and s6 matches the options given.

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17 Dec 2024, 03:50

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let's do this one the lazy way: let the terms s6and s4 both be not div by 4... we then get a difference of 6, only 1 possible solution: 16,22.

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17 Dec 2024, 04:02

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It is easy to go from choices given. Let's consider S4 is 5, it is not divisible by 4 that gives S5 is 8. S5 is divisible by 4, so it gives S6 is 10. But there is no choice for 10.

Same way by checking S4 is 8 ---> S5 is 10 ---> S6 is 13. We have both the numbers in the columns.

So the answers are s4 = 8 and s6 = 13

(OR)

Alternative way is there is only one term between S4 and S6. So, the possible difference between s4 and s6 is 5 (when one of s4 or s5 value is divisible by 4) or 6 (when s4 and s5 are not divisible by 4).

So let's check with difference is 5. Only 8 and 13 pair gives the difference is 5. So those are the correct answers.

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17 Dec 2024, 04:38

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Quote:

It's better to test the answer, starting from the lower number $s_4$
If $s_4$ = 8, it is divisible by 4
then $s_5$ = $s_4$+2 = 8+2 = 10

Since $s_5$=10 and is not divisible by 4,
then $s_6$= $s_5$+3 = 10+3 = 13

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17 Dec 2024, 06:29

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without loss of generality, suppose S0 is divisible by 4.
S1= S0+2
S2 = S1+3 = S0 + 5, since S1 is not divisible by 4
S3 = S2+3 = S0 + 8, since S2 is not divisible by 4
S4 = S3+2 = S0 + 10, since S3 = S0 + 8, which is divisible by 4
S5 = S4+3 = S0 + 13, since S4 = S0 + 10, which is not divisible by 4
S6 = S5+3 = S0 + 16, since S5 = S0 + 13, which is not divisible by 4

S6 - S4 = S0 + 16 - S0 + 10 = 6
Only two values from the options match, where the difference between two terms is 6 - 16,22.

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17 Dec 2024, 07:04

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Hit and trial method -

If S4 is 5, then it will follow the second equation for S5 as 5 is not divisible by 4.
So, S5 = S4 + 3 = 5+3 = 8.
Since S5 = 8 is divisible by 4, it will follow equation 1 for S6.
S6 = S5 + 2 = 8 + 2 = 10.

10 is not in the options, hence, not correct.

If S4 is 8, then, for S5, it will follow equation 1 as 8 is divisible by 4.
So, S5 = S4 + 2 = 8+2 = 10.
Since S5 is not divisible by 4, it will follow equation 2 for S6.

S6 = S5 + 3 = 10 + 3 = 13.

S4 = 8, S6 = 13.

Bunuel

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17 Dec 2024, 07:50

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Lets note that the sequence $s_n, s_{n+1}$ is always increasing no matter whether s_n is divisbile by 4 or not.
Now, lets plug and chuck values for s_4 in the ascending order of options.
After plugging 5 for s_4 gives 10 for s_6 ;
plugging 8 for s_4 gives 13 for s_6. Hence (8,13) pair is consistent with the criterion set. Hence is the right option !

Bunuel

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17 Dec 2024, 07:51

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Bunuel

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This kind of question we need to go by option. So let's pick s4 first and then see if corresponding s6 is present or not.

s4 =5, s5 = 8, s6 = 10. (Not avb)
s4=8, s5=10, s6 =13. (Avb)

Hence s4 = 8, s6 =13

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17 Dec 2024, 07:52

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From the above question, we observe a pattern in the numbers in the series

lets say the number x is divisible by 4,the next number is x+2 which isnt divisible by 4, then it gets converted to x+5, which is not divisible by 4, then it gets converted to x+8 which is divisble by 4, and repeat

So the series is x, x+2, x+5, x+8, x+10,....

So from that we can see,
if we substitute s4 as 8 then s6 would be 13

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17 Dec 2024, 08:48

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Bunuel

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The best thing to understand is to ascertain that the values are ascending i.e. increasing from S1 to S2 to S3 and so on.
Which means that between s4 to s6 as well the values increase.

Now, trial and error worked best for me. So,
I. if S4 = 5 then S5=8 (s4 not divisible by 4, so add 3) and S6=10 (s5 divisible by 4, so add 2)
II. if S4 = 8 then S5=10 (s4 divisible by 4, so add 2) and S6=13 (s5 not divisible by 4, so add 3) Jackppot.

If you try this with others, Option B and C for I and II respectively will still stand. Trust yourself and avoid redundancies.

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17 Dec 2024, 08:54

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Hii,
Lets take S4=8
as it is divisible by 4 then S5 =10
as S5 is not divisible by 4 so
S6 will be 13
so pair will be S4 =8
S6=13
Thanks

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18 Dec 2024, 08:23

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S4= 8
8 is divisible by 4, so S5=10
10 is not divisible by 4, so S6=13

(S4,S6) = (8,13)

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21 Dec 2024, 09:47

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Analyzing the conditions given,
If $s_n$ is divisible by 4, the difference between consecutive terms is 2
If $s_n$ is not divisible by 4, the difference between consecutive terms is 3

Let $s_4$ be 8 (divisible by 4),
$s_5 = s_4 + 2 = 8 + 2 = 10$
$s_6 = s_5 + 3 = 10 + 3 = 13$

Now let $s_4$ be 7 (not divisible by 4),
$s_5 = s_4 + 3 = 7 + 3 = 10$
$s_6 = s_5 + 3 = 10 + 3 = 13$

Answer:
$s_4$ = 8
$s_6$ = 13