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Consider a function f satisfying f (x + y) = f (x) f (y) where x,y are positive integers, and f(1) = 2. If f(a + 1) +f (a + 2) +... + f(a + n) = 16 (2n -1) then a is equal to
Correct answer is '3'. Can you explain this answer?
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Verified Answer
Consider a function f satisfying f (x + y) = f (x) f (y) where x,y are...
f(x + y)=f(x)f(y)
Hence, f(2) = f(1+1)=f(1)*f(1)=2*2=4
f(3)=f(2+1)=f(2)*f(1)=4*2=8
f(4)=f(3+1)=f(3)*f(1)=8*2=16
……… =>f(x)=2x
Now,f(a + 1)+f(a + 2) + ... + f(a + n) = 16(2n-1)
On putting n=1 in the equation we get, f(a+1)=16
⇒ f(a)*f(1)=16 (It is given that f (x + y) = f (x) f (y))
⇒ 2a*2=16
⇒ a=3
Most Upvoted Answer
Consider a function f satisfying f (x + y) = f (x) f (y) where x,y are...
f(x + y)=f(x)f(y)
Hence, f(2) = f(1+1)=f(1)*f(1)=2*2=4
f(3)=f(2+1)=f(2)*f(1)=4*2=8
f(4)=f(3+1)=f(3)*f(1)=8*2=16
……… =>f(x)=2x
Now,f(a + 1)+f(a + 2) + ... + f(a + n) = 16(2n-1)
On putting n=1 in the equation we get, f(a+1)=16
⇒ f(a)*f(1)=16 (It is given that f (x + y) = f (x) f (y))
⇒ 2a*2=16
⇒ a=3
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Consider a function f satisfying f (x + y) = f (x) f (y) where x,y are...
Understanding the given function and condition:
The given function f(x + y) = f(x) f(y) for positive integers x and y indicates that the function is multiplicative.
Given f(1) = 2, we can see that f(2) = f(1) f(1) = 2^2 = 4, f(3) = f(1) f(2) = 2*4 = 8, and so on.
Using the given condition:
We are given that f(a + 1) + f(a + 2) + ... + f(a + n) = 16(2n - 1).
Substituting the values of f(x) from the function, we get 2^a + 2^(a+1) + ... + 2^(a+n-1) = 16(2n - 1).
Simplifying the expression:
2^a(1 + 2 + ... + 2^(n-1)) = 16(2n - 1)
2^a(2^n - 1) = 16(2n - 1)
2^(a+n) - 2^a = 16(2n - 1)
Finding the value of a:
Let's try different values of a to satisfy the equation.
For a = 1, we get 2^(1+n) - 2 = 16(2n - 1), which does not hold true.
For a = 2, we get 2^(2+n) - 4 = 16(2n - 1), which also does not hold true.
For a = 3, we get 2^(3+n) - 8 = 16(2n - 1), which holds true.
Therefore, the correct value of a is 3.
The given function f(x + y) = f(x) f(y) for positive integers x and y indicates that the function is multiplicative.
Given f(1) = 2, we can see that f(2) = f(1) f(1) = 2^2 = 4, f(3) = f(1) f(2) = 2*4 = 8, and so on.
Using the given condition:
We are given that f(a + 1) + f(a + 2) + ... + f(a + n) = 16(2n - 1).
Substituting the values of f(x) from the function, we get 2^a + 2^(a+1) + ... + 2^(a+n-1) = 16(2n - 1).
Simplifying the expression:
2^a(1 + 2 + ... + 2^(n-1)) = 16(2n - 1)
2^a(2^n - 1) = 16(2n - 1)
2^(a+n) - 2^a = 16(2n - 1)
Finding the value of a:
Let's try different values of a to satisfy the equation.
For a = 1, we get 2^(1+n) - 2 = 16(2n - 1), which does not hold true.
For a = 2, we get 2^(2+n) - 4 = 16(2n - 1), which also does not hold true.
For a = 3, we get 2^(3+n) - 8 = 16(2n - 1), which holds true.
Therefore, the correct value of a is 3.
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Consider a function f satisfying f (x + y) = f (x) f (y) where x,y are positive integers, and f(1) = 2. If f(a + 1) +f (a + 2) +... + f(a + n) = 16 (2n -1) then a is equal toCorrect answer is '3'. Can you explain this answer?
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Consider a function f satisfying f (x + y) = f (x) f (y) where x,y are positive integers, and f(1) = 2. If f(a + 1) +f (a + 2) +... + f(a + n) = 16 (2n -1) then a is equal toCorrect answer is '3'. Can you explain this answer? for SSC 2024 is part of SSC preparation. The Question and answers have been prepared according to the SSC exam syllabus. Information about Consider a function f satisfying f (x + y) = f (x) f (y) where x,y are positive integers, and f(1) = 2. If f(a + 1) +f (a + 2) +... + f(a + n) = 16 (2n -1) then a is equal toCorrect answer is '3'. Can you explain this answer? covers all topics & solutions for SSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a function f satisfying f (x + y) = f (x) f (y) where x,y are positive integers, and f(1) = 2. If f(a + 1) +f (a + 2) +... + f(a + n) = 16 (2n -1) then a is equal toCorrect answer is '3'. Can you explain this answer?.
Consider a function f satisfying f (x + y) = f (x) f (y) where x,y are positive integers, and f(1) = 2. If f(a + 1) +f (a + 2) +... + f(a + n) = 16 (2n -1) then a is equal toCorrect answer is '3'. Can you explain this answer? for SSC 2024 is part of SSC preparation. The Question and answers have been prepared according to the SSC exam syllabus. Information about Consider a function f satisfying f (x + y) = f (x) f (y) where x,y are positive integers, and f(1) = 2. If f(a + 1) +f (a + 2) +... + f(a + n) = 16 (2n -1) then a is equal toCorrect answer is '3'. Can you explain this answer? covers all topics & solutions for SSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a function f satisfying f (x + y) = f (x) f (y) where x,y are positive integers, and f(1) = 2. If f(a + 1) +f (a + 2) +... + f(a + n) = 16 (2n -1) then a is equal toCorrect answer is '3'. Can you explain this answer?.
Solutions for Consider a function f satisfying f (x + y) = f (x) f (y) where x,y are positive integers, and f(1) = 2. If f(a + 1) +f (a + 2) +... + f(a + n) = 16 (2n -1) then a is equal toCorrect answer is '3'. Can you explain this answer? in English & in Hindi are available as part of our courses for SSC.
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