If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = 91, f(15) = 617, then f(10) equals

(2018)

a)
55

b)
54

c)
52

d)
51

Correct answer is option 'B'. Can you explain this answer?

Question

If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = 91, f(15) = 617, then f(10) equals(2018)a)55b)54c)52d)51Correct answer is option 'B'. Can you explain this answer?

Answer

Calculation:

⇒ 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)

By solving,

⇒ f(12) + f(13) = 617 - f(13) [ using (1) and (2) ]

⇒ 2f(11) + 3f(12) = 617 [ usining (3) ]

⇒ f(12) = 145

⇒ f(12) = f(11) + f(10)

∴ f(10) = 54

Answer

f(x + 2) = f(x) + f(x + 1)
f(11) = 91
Let f(12) = a
f(13) = 91 + a
f(14) = 91 + 2a
f(15) = 182 + 3a = 617
⇒ a = 145
Now, f(10) = f(12) – f(11) = 145 – 91 = 54

Answer

Given information:
- f(x + 2) = f(x) + f(x + 1) for all positive integers x
- f(11) = 91
- f(15) = 617

To find:
f(10)

Approach:
We can use the given information and the given equation to find the value of f(10). We will start by finding the values of f(12), f(13), and f(14) using the given equation. Then, we will use these values to find f(10).

Solution:
Let's find the values of f(12), f(13), and f(14) using the given equation:
- f(12) = f(10) + f(11)
- f(13) = f(11) + f(12)
- f(14) = f(12) + f(13)

We know that f(11) = 91, so we can substitute this value in the above equations:
- f(12) = f(10) + 91
- f(13) = 91 + f(12)
- f(14) = f(12) + f(13)

Now, let's substitute the value of f(12) in the equations for f(13) and f(14):
- f(13) = 91 + (f(10) + 91) = 2f(10) + 182
- f(14) = (f(10) + 91) + (2f(10) + 182) = 3f(10) + 273

We are given that f(15) = 617, so we can substitute this value in the above equation:
617 = 3f(10) + 273

Now, let's solve this equation to find the value of f(10):
3f(10) = 617 - 273
3f(10) = 344
f(10) = 344/3

Since f(10) is a positive integer, the value of f(10) cannot be 344/3. Therefore, the given information is inconsistent.

Conclusion:
The given information is inconsistent. Therefore, it is not possible to determine the exact value of f(10) based on the given information. The correct answer would be 'Cannot be determined'. Option B (54) is incorrect.

Answer

f(x + 2) = f(x) + f(x + 1)
f(11) = 91
Let f(12) = a
f(13) = 91 + a
f(14) = 91 + 2a
f(15) = 182 + 3a = 617
⇒ a = 145
Now, f(10) = f(12) – f(11) = 145 – 91 = 54

Answer

f(x + 2) = f(x) + f(x + 1)
f(11) = 91
Let f(12) = a
f(13) = 91 + a
f(14) = 91 + 2a
f(15) = 182 + 3a = 617
⇒ a = 145
Now, f(10) = f(12) – f(11) = 145 – 91 = 54

Similar CAT Doubts

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