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If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) =

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Bunuel
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If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = [#permalink] If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) =   03 Apr 2020, 05:58
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If \(f(x + 2) = f(x) + f(x + 1)\) for all positive integers x, and \(f(11) = 91\), \(f(15) = 617\), then what is the value of \(f(10)\) ?

A. 53
B. 54
C. 55
D. 56
E. 57


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BrentGMATPrepNow
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Re: If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = [#permalink] If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) =   03 Apr 2020, 06:23
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Bunuel wrote:
If \(f(x + 2) = f(x) + f(x + 1)\) for all positive integers x, and \(f(11) = 91\), \(f(15) = 617\), then what is the value of \(f(10)\) ?

A. 53
B. 54
C. 55
D. 56
E. 57


Are You Up For the Challenge: 700 Level Questions


Given: f(x + 2) = f(x) + f(x + 1)
This means: f(12) = f(10) + f(11)
So, since f(13) = f(11) + f(12), we can write: f(13) = f(11) + f(12)
Replace f(12) to get: f(13) = f(11) + f(10) + f(11)
Simplify: f(13) = 2f(11) + f(10)

Likewise, take: f(14) = f(12) + f(13)
Substitute: f(14) = f(10) + f(11) + 2f(11) + f(10)
Simplify: f(14) = 3f(11) + 2f(10)

Finally, since f(15) = f(13) + f(14), ...
...we can write: f(15) = 2f(11) + f(10) + 3f(11) + 2f(10)
Simplify: f(15) = 5f(11) + 3f(10)
Substitute values to get: 617 = 5(91) + 3f(10)
Evaluate to get: 617 = 455 + 3f(10)
Subtract 455 from both sides: 162 = 3f(10)
Divide both sides by 3 to get: 54 = f(10)

Answer: B

Cheers,
Brent
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Re: If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = [#permalink] If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) =   03 Apr 2020, 07:04
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Bunuel wrote:
If \(f(x + 2) = f(x) + f(x + 1)\) for all positive integers x, and \(f(11) = 91\), \(f(15) = 617\), then what is the value of \(f(10)\) ?

A. 53
B. 54
C. 55
D. 56
E. 57


\(f(15) = 617\); \(f(15) = f(13)+f(14)\)
\(f(13) = f(11)+f(12)\)
\(f(12) = f(10)+f(11)\)
\(f(14) = f(12)+f(13)\)
\(f(15) = [f(11)+f(10)+f(11)]+[f(10)+f(11)+f(11)+f(10)+f(11)]\)
\(617 = 5*f(11)+3*f(10)\)
\(3*f(10)= 617 -5*91\)
\(f(10)= \frac{162}{3}=54\)
Ans: B
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Re: If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = [#permalink] If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) =   04 Apr 2020, 09:15
Given , f(x + 2) = f(x) + f(x + 1)
f(15) = f(13) + f(14)
f(13) + f(14) = 617 —————1)
f(12) + f(13) = f(14) ————-2)
f(11) + f(12) = f(13)————-3)
from eq 1) , 2) & 3)
2f(11) + 3f(12) = 617
Putting f(11) = 91
f(12) = (617 – 2*91)/3 = 145
f(12) = f(11) + f(10)
so f(10) = f(12) – f(11) = 145 – 91 =54
Answer: B
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Re: If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = [#permalink] If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) =   04 May 2020, 09:12
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Bunuel wrote:
If \(f(x + 2) = f(x) + f(x + 1)\) for all positive integers x, and \(f(11) = 91\), \(f(15) = 617\), then what is the value of \(f(10)\) ?

A. 53
B. 54
C. 55
D. 56
E. 57


617=f14+f13=f13+f12+f13=2(f13)+f12=2(f12+f11)+f12
617=3f12+91, f12=435/3=145, f12=f11+f10
145=91+f10, f10=54

ans (B)
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Re: If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = [#permalink] If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) =   06 Jan 2022, 11:44
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Given that \(f(x + 2) = f(x) + f(x + 1)\) for all positive integers x , and \(f(11) = 91\), \(f(15) = 617\) and we need to find the value of \(f(10)\)

\(f(x + 2) = f(x) + f(x + 1)\) - this means that f(any positive integer) = sum of f of previous two numbers
=> f(15) = f(13) + f(14) ...(1)
f(14) = f(12) + f(13) ...(2)
f(13) = f(11) + f(12) ....(3)
f(12) = f(10) + f(11)

Putting value of f(12) in (3) we get
f(13) = f(11) + f(10) + f(11) = f(10) + 2*f(11)

Putting value of f(13) and f(12) in (2) we get
f(14) = f(10) + f(11) + f(10) + 2*f(11) = 2*f(10) + 3*f(11)

Putting value of f(14) and f(13) in (1) we get
f(15) = f(10) + 2*f(11) + 2*f(10) + 3*f(11) = 3*f(10) + 5*f(11)
=> f(10) = \(\frac{f(15) - 5*f(11) }{3}\) = \(\frac{617 - 5*91}{3}\) = \(\frac{617-455}{3}\) = \(\frac{162}{3}\) = 54

So, Answer will be B
Hope it helps!

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Re: If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = [#permalink] If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) =   04 Jul 2024, 05:32
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Re: If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = [#permalink]
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