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If 'f' is the real function satisfying 2f(xy) = [f(x)]y + [f(y)]x for all real values of x, y and f(1) = 2, then f(1) +f(2)+f(3) + ... +f(10) is equal to
Correct answer is '2046'. Can you explain this answer?
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If 'f' is the real function satisfying 2f(xy) = [f(x)]y+ [f(y)...
The above series is in G.R with ratio = 2 and initial term = 2
Answer: 2046
Most Upvoted Answer
If 'f' is the real function satisfying 2f(xy) = [f(x)]y+ [f(y)...
Given Information:
The real function f satisfies the equation 2f(xy) = [f(x)]y [f(y)]x for all real values of x and y. It is also given that f(1) = 2.
To Find:
We need to find the value of f(1) + f(2) + f(3) + ... + f(10).
Solution:
Step 1: Finding the value of f(2)
To find the value of f(2), we substitute x = 1 and y = 2 into the given equation:
2f(1 * 2) = [f(1)]2 [f(2)]1
2f(2) = 22 [f(2)]1
2f(2) = 4 + f(2)
f(2) = 4
Step 2: Finding the value of f(3)
To find the value of f(3), we substitute x = 1 and y = 3 into the given equation:
2f(1 * 3) = [f(1)]3 [f(3)]1
2f(3) = 23 [f(3)]1
2f(3) = 8 + f(3)
f(3) = 8
Step 3: Finding the value of f(4)
To find the value of f(4), we substitute x = 2 and y = 2 into the given equation:
2f(2 * 2) = [f(2)]2 [f(2)]2
2f(4) = 42 [f(4)]2
2f(4) = 16 + f(4)
f(4) = 16
Step 4: Finding the value of f(5)
To find the value of f(5), we substitute x = 1 and y = 5 into the given equation:
2f(1 * 5) = [f(1)]5 [f(5)]1
2f(5) = 25 [f(5)]1
2f(5) = 32 + f(5)
f(5) = 32
Step 5: Finding the value of f(6)
To find the value of f(6), we substitute x = 2 and y = 3 into the given equation:
2f(2 * 3) = [f(2)]3 [f(3)]2
2f(6) = 43 [f(3)]2
2f(6) = 64 + 8
f(6) = 36
Step 6: Finding the value of f(7)
To find the value of f(7), we substitute x = 1 and y = 7 into the given equation:
2f(1 * 7) = [f(1)]7 [f(7)]1
2f(7) = 27 [f(7)]1
2f(7) = 128 + f(7)
f(7) = 128
The real function f satisfies the equation 2f(xy) = [f(x)]y [f(y)]x for all real values of x and y. It is also given that f(1) = 2.
To Find:
We need to find the value of f(1) + f(2) + f(3) + ... + f(10).
Solution:
Step 1: Finding the value of f(2)
To find the value of f(2), we substitute x = 1 and y = 2 into the given equation:
2f(1 * 2) = [f(1)]2 [f(2)]1
2f(2) = 22 [f(2)]1
2f(2) = 4 + f(2)
f(2) = 4
Step 2: Finding the value of f(3)
To find the value of f(3), we substitute x = 1 and y = 3 into the given equation:
2f(1 * 3) = [f(1)]3 [f(3)]1
2f(3) = 23 [f(3)]1
2f(3) = 8 + f(3)
f(3) = 8
Step 3: Finding the value of f(4)
To find the value of f(4), we substitute x = 2 and y = 2 into the given equation:
2f(2 * 2) = [f(2)]2 [f(2)]2
2f(4) = 42 [f(4)]2
2f(4) = 16 + f(4)
f(4) = 16
Step 4: Finding the value of f(5)
To find the value of f(5), we substitute x = 1 and y = 5 into the given equation:
2f(1 * 5) = [f(1)]5 [f(5)]1
2f(5) = 25 [f(5)]1
2f(5) = 32 + f(5)
f(5) = 32
Step 5: Finding the value of f(6)
To find the value of f(6), we substitute x = 2 and y = 3 into the given equation:
2f(2 * 3) = [f(2)]3 [f(3)]2
2f(6) = 43 [f(3)]2
2f(6) = 64 + 8
f(6) = 36
Step 6: Finding the value of f(7)
To find the value of f(7), we substitute x = 1 and y = 7 into the given equation:
2f(1 * 7) = [f(1)]7 [f(7)]1
2f(7) = 27 [f(7)]1
2f(7) = 128 + f(7)
f(7) = 128
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If 'f' is the real function satisfying 2f(xy) = [f(x)]y+ [f(y)]xfor all real values of x, y and f(1) = 2, then f(1) +f(2)+f(3) + ... +f(10) is equal toCorrect answer is '2046'. Can you explain this answer?
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If 'f' is the real function satisfying 2f(xy) = [f(x)]y+ [f(y)]xfor all real values of x, y and f(1) = 2, then f(1) +f(2)+f(3) + ... +f(10) is equal toCorrect answer is '2046'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about If 'f' is the real function satisfying 2f(xy) = [f(x)]y+ [f(y)]xfor all real values of x, y and f(1) = 2, then f(1) +f(2)+f(3) + ... +f(10) is equal toCorrect answer is '2046'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 'f' is the real function satisfying 2f(xy) = [f(x)]y+ [f(y)]xfor all real values of x, y and f(1) = 2, then f(1) +f(2)+f(3) + ... +f(10) is equal toCorrect answer is '2046'. Can you explain this answer?.
If 'f' is the real function satisfying 2f(xy) = [f(x)]y+ [f(y)]xfor all real values of x, y and f(1) = 2, then f(1) +f(2)+f(3) + ... +f(10) is equal toCorrect answer is '2046'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about If 'f' is the real function satisfying 2f(xy) = [f(x)]y+ [f(y)]xfor all real values of x, y and f(1) = 2, then f(1) +f(2)+f(3) + ... +f(10) is equal toCorrect answer is '2046'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 'f' is the real function satisfying 2f(xy) = [f(x)]y+ [f(y)]xfor all real values of x, y and f(1) = 2, then f(1) +f(2)+f(3) + ... +f(10) is equal toCorrect answer is '2046'. Can you explain this answer?.
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