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emmainna [20.7K]

2 years ago

15

Evie and her father are comparing their ages. At the current time, Evie's father is 36 years older than her. Theee years from no

w, Evie's father will be five times her age at that point. How old is Evie now?

1 answer:

artcher [175]2 years ago

7 0

Let F be the father's current age, and E be Evie's current age.
F = E + 36 this is the current relation between the ages
F + 3 = 5(E + 3) this the relationship in 3 year (hence the + 3)
Then solve by substituting one equation into another:
(E + 36) + 3 = 5(E + 3)
E + 39 =  5E + 15
  24 = 4E
  6 = E
Evie's current age is 6.

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Answer with step-by-step explanation:

We are given that the recurrence relation

$f_n=5f_{n-4}+3f_{n-5}$

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Initial condition

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The recurrence relation of Fibonacci numbers

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Apply this

$f_n=(f_{n-2}+f_{n-3})+f_{n-2}=2f_{n-2}+f_{n-3}$

$f_n=2(f_{n-3}+f_{n-4})+f_{n-3}=3f_{n-3}+2f_{n-4}$

$f_n=3(f_{n-4}+f_{n-5})+2f_{n-4}=5f_{n-4}+3f_{n-5}$

Substitute n=2

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$f_3=f_2+f_1=1+1=2$

$f_4=f_3+f_2=2+1=3$

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Now, we have to show that $f_{5n}$ is divisible by 5 for n=1,2,3,..

Now replace n by 5n

$f_{5n}=5f_{5n-4}+3f_{5n-5}$

Apply induction

Substitute n=1

$f_5=5f_1+3f_0=5+0=5$

It is true for n=1

Suppose it is true for n=k

$f_{5k}=5f_{5k-4}+3f_{5k-5}$ is divisible 5

Let $f_{5k}=5q$

Now, we shall prove that for n=k+1 is true

$f_{5k+5}=5f_{5k+5-4}+3f_{5k+5-5}=5f_{5k+1}+3f_{5k}=5f_{5k+1}+3(5q)$

$f_{5k+5}=5(f_{5k+1}+3q)$

It is multiple of 5 .Therefore, it is divisible by 5.

It is true for n=k+1

Hence, the $f_{5n}$ is divisible by 5 for n=1,2,3,..

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<em><u>Solution:</u></em>

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