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12 Days of Christmas GMAT Competition 2025 - Day 7: If f(n) = |3^n - 2

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Bunuel

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If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

M46-19

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Bunuel

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Bunuel

If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

GMAT Club Official Explanation:

f(n) = |3^n - 27^15 - 81^2|

f(n) = |3^n - (3^3)^15 - (3^4)^2|

f(n) = |3^n - 3^45 - 3^8|

f(n) = |3^n - 3^8(3^37 + 1)|

Note that 3^37 + 1 is very close to 3^37, as adding 1 has a negligible effect on such large numbers. Therefore, we can approximate:

f(n) = |3^n - 3^8(3^37 + 1)| ≈

≈ |3^n - 3^8*3^37| =

= |3^n - 3^45|

The smallest possible value of an absolute value expression is 0, which in the approximation above will be obtained when n = 45.

Answer: C.

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23 Dec 2025, 09:32

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If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

given
f(n) = l3^n-3^45-3^8l
smallest possible value of n will be at
f(n) = l3^n-(3^45+3^8)l
when n is 45
f(n) = 3^8
when n is 46
f(n) = 2*3^45-3^8

OPTION C 45 is correct

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Bunuel

If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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|3^n - 3^45 - 3^8|
it is lowest when 3^n is closest to 3^45 + 3^8
this is lowest when n = 45
ans is 45

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f(n)= |3^n -27^15 -81^2| = |3^n -3^45 -3^8|
The value of f(n) is minimised when 3^n is close to sum of (3^45 +3^8). Lets try n=45 and n=46

C. n=45 ; |3^45 -3^45 -3^8| = |(3^45)(1-1)-3^8| = 3^8 =6,561
D. n=46; |3^46 -3^45 -3^8)| = |(3^45)(3-1) -3^8)| = |(3^45 (2) -3^8)| ....It is bigger than f(45)

C

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Simplifying the equation:
|3^n - 27^15 - 81^2|
|3^n - 3^45 - 3^8|
|3^8 {3^(n-8) - 3^37 - 1}|

For the minimum value of above expression, value of {3^(n-8) - 3^37 - 1} should be minimum which when multiplied with 3^8 would give lowest value. For this to happen
n-8 = 38
n = 45
This will give the value of the {} bracket as -1.

Hence, Answer is Option C

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f(n) = |3^n - 27^15 - 81^2| = |3^n - (3^3)^15 - (3^4)^2| = |3^n - 3^45 - 3^8|

A. f(15) = |3^15 - 3^45 - 3^8| = 3^45 + 3^8 - 3^15 = 3^45 approx
B. f(44) = |3^44 - 3^45 - 3^8| = |3^44(-2) - 3^8| = 2*3^44 + 3^8 = 2*3^44 approx
C. f(45) = |3^45 - 3^45 - 3^8| = 3^8
D. f(46) = |3^46 - 3^45 - 3^8| = |3^45(2) - 3^8| = 2^3^45 - 3^8 = 2*3^45 approx
E. f(49) = |3^49 - 3^45 - 3^8| = |3^45(3^4-1) - 3^8| = 80*3^45 - 3^8 = 80*3^45 approx

n = 45; f(45) = 3^8 results in smallest possible value of f(n)

IMO C

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f(n) = |3^n - 3^45 - 3^8|

f(n) is smallest when the result is 0 or closest to it

3^n = 3^45 + 3^8
3^n = 3^45 (1 + 3^-37)
3^-37 is pretty small number compared to 3^45, so approximately the LHS = 3^45
3^n = 3^45
n=45

Answer C

Bunuel

If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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Bunuel

If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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Lets rewrite everything as power of 3:
27^15 = (3^3)^15= 3^45
81^2 = (3^4)^2 = 3^8

f(n) = |3^n - 3^45 - 3^8| = |3^n - (3^45 + 3^8)|
3^45 >> 3^8
so 3^45 + 3^8 = 3^45(1 + 3^(8-45)) = 3^45(1 + 3^-37)
since 3^-37 is extremely small, 3^45 + 3^8 is just a bit larger than 3^45 but still much closer to 3^45

thus n = 45

Option C

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|x|> 0. Minimum value |x|=0 which means x=0

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In this equation on solving will become

f(n) = |3^n - 3^45 -3^8|

Now if we check each option then
only if we take n = 45
then we can least value of 3^8
otherwise it will be all more than this
for example since subtracting
3^45 - 3^44 = 2* 3^44

therefore,
answer is 45
C

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option c is correct
f(n)=|3^n - 27^15- 81^2 |
n=45
=|3^45 - 3^45 - 3^8|
=3^8

answer is 45

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f(n) = |3^n - 27^15 -81^2|
=|3^n - 3^45 - 3^8|

if n=45, f(n) = 3^8

if we consider other values of n, then 3^8 will come out as common factor, and some more product terms will be present. The value will be more than 3^8.
n=45

Ans C

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first rewrite f(n) as f(n) = | 3^n - (3^45 + 3^8)|

to make f(n) as small as possible, 3^n must be as close as possible to (3^45 + 3^8)

you can intuitively eliminate option A and E (the resulting f(n) would be a huge number because 3^n would be much smaller/bigger than 3^45 + 3^8)

if n = 44 -> f(n) = |3^44 - 3^45 - 3^8 | = 3^44*2 - 3^8

if n = 45 -> f(n) = 3^8

if n = 46 -> f(n) = |3^46 - 3^45 - 3^8 | = 3^45*2 - 3^8

clearly with n=45, f(n) is smaller.

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Further simplification of the equation gives us:

f(n) = | $3^n$ - $3^45$ - $3^8$ |
f(n) = | $3^n$ - ( $3^45$ + $3^8$ )
Thus, in order to minimise f(n), we want a number as close as possible to ($3^45$ + $3^8$)
Among the options, this number is clearly when n=45.

Bunuel

If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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f(n) = |3^n-3^45-3^8| or |3^n - 3^8(3^37 -1)|

To get a lower value of f(n), 3^n and the second term should be as close as possible.

Given that 3^37 -1 is almost equal to 3^37, 3^n should be close or equal to 3^8(3^37) or 3^45

It will be as close as possible when n =45

Therefore, Option C

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given f(n)=|3^n - 27^15 - 81^2| need to find n which will result in the smallest value of f(n)?
f(n)=| 3^n - 3^45 - 3^8|
hence its n=45 that will result in the smallest value of f(n)=81^2 or 3^8 as the first two terms cancel out.
rest all values of n will result in a much higher value of f(n).
so C

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Bunuel

If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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Express in base of 3 = |3ˆn - 3ˆ45 - 3ˆ8|
Since it's an absolute function, to get minimum value we need to eliminate the variables to zero. Getting the lowest value of n will not decrease the value of function. Thus putting n=45, leaves us with 3ˆ8.
Even if a lower value is chosen say 44, we will get |3ˆ44(1-3) - 3ˆ8| or |(-2)(3ˆ44) - 3ˆ8| which is higher than 3ˆ8 .

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23 Dec 2025, 14:42

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|3^n - 3^45 - 3^8 |
we can directly test the num from choice
since we have 3^45 in our equation and in choice, i will start with that.
that will give me |-(3^8)| = 6561
any other number will be greater than this because we wont be able to cancel out any term like earlier one.
so every value of n will have larger value than the above one.
ans is C=45

Bunuel

If f(n) = |3^n - 27^15 - 81^2|, which value of n results in the smallest possible value for f(n)?

A. 15
B. 44
C. 45
D. 46
E. 49

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f(n) = |3^n -3^45-3^8|

I tries iteratively, Clearly all other options except (C) are giving higher value as it have higher output value due to high powers.

Hence (C) is the answer. It is only giving 3^8 as function output, which is less in comparison.