Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > The recurrence equationT(1) = 1T(n) = 2T(n - ...
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The recurrence equation
T(1) = 1
T(n) = 2T(n - 1) + n, n ≥ 2
T(n) = 2T(n - 1) + n, n ≥ 2
evaluates to
- a)2n + 1- n - 2
- b)2n - n
- c)2n + 1 - 2n - 2
- d)2n - n
Correct answer is option 'A'. Can you explain this answer?
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The recurrence equationT(1) = 1T(n) = 2T(n - 1) + n, n ≥ 2evalua...
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The recurrence equationT(1) = 1T(n) = 2T(n - 1) + n, n ≥ 2evalua...
Recurrence Equation
The given recurrence equation is:
T(1) = 1
T(n) = 2T(n - 1) for n ≥ 2
This equation defines the value of the function T for different values of n. The function T represents the time complexity of some algorithm or process.
Solving the Recurrence Equation
To solve this recurrence equation and find the value of T(n), we can use a technique called substitution method.
Step 1: Start by substituting the base case value:
T(1) = 1
Step 2: Substitute the value of T(n - 1) in terms of T(n - 2), and continue this process until we reach the base case:
T(n) = 2T(n - 1)
= 2(2T(n - 2))
= 2^2T(n - 2)
= 2^3T(n - 3)
= ...
= 2^(n-1)T(1)
Step 3: Substitute the base case value:
T(n) = 2^(n-1)T(1)
= 2^(n-1) * 1
= 2^(n-1)
Final Solution
Therefore, the solution to the given recurrence equation T(n) = 2T(n - 1) is:
T(n) = 2^(n-1)
Answer Explanation:
Now let's compare the options given:
a) 2n - 1 - n - 2
b) 2n - n
c) 2n - 1 - 2n - 2
d) 2n - n
Comparing the solution we obtained, 2^(n-1), with the options:
a) 2n - 1 - n - 2 = n - 3 (not equal to 2^(n-1))
b) 2n - n = n (not equal to 2^(n-1))
c) 2n - 1 - 2n - 2 = -n - 3 (not equal to 2^(n-1))
d) 2n - n = n (not equal to 2^(n-1))
None of the given options match the solution we obtained. Therefore, it seems that the correct answer option is not mentioned in the given options.
The given recurrence equation is:
T(1) = 1
T(n) = 2T(n - 1) for n ≥ 2
This equation defines the value of the function T for different values of n. The function T represents the time complexity of some algorithm or process.
Solving the Recurrence Equation
To solve this recurrence equation and find the value of T(n), we can use a technique called substitution method.
Step 1: Start by substituting the base case value:
T(1) = 1
Step 2: Substitute the value of T(n - 1) in terms of T(n - 2), and continue this process until we reach the base case:
T(n) = 2T(n - 1)
= 2(2T(n - 2))
= 2^2T(n - 2)
= 2^3T(n - 3)
= ...
= 2^(n-1)T(1)
Step 3: Substitute the base case value:
T(n) = 2^(n-1)T(1)
= 2^(n-1) * 1
= 2^(n-1)
Final Solution
Therefore, the solution to the given recurrence equation T(n) = 2T(n - 1) is:
T(n) = 2^(n-1)
Answer Explanation:
Now let's compare the options given:
a) 2n - 1 - n - 2
b) 2n - n
c) 2n - 1 - 2n - 2
d) 2n - n
Comparing the solution we obtained, 2^(n-1), with the options:
a) 2n - 1 - n - 2 = n - 3 (not equal to 2^(n-1))
b) 2n - n = n (not equal to 2^(n-1))
c) 2n - 1 - 2n - 2 = -n - 3 (not equal to 2^(n-1))
d) 2n - n = n (not equal to 2^(n-1))
None of the given options match the solution we obtained. Therefore, it seems that the correct answer option is not mentioned in the given options.
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Question Description
The recurrence equationT(1) = 1T(n) = 2T(n - 1) + n, n ≥ 2evaluates toa)2n + 1- n - 2b)2n- nc)2n + 1- 2n - 2d)2n- nCorrect answer is option 'A'. Can you explain this answer? for Computer Science Engineering (CSE) 2025 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about The recurrence equationT(1) = 1T(n) = 2T(n - 1) + n, n ≥ 2evaluates toa)2n + 1- n - 2b)2n- nc)2n + 1- 2n - 2d)2n- nCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The recurrence equationT(1) = 1T(n) = 2T(n - 1) + n, n ≥ 2evaluates toa)2n + 1- n - 2b)2n- nc)2n + 1- 2n - 2d)2n- nCorrect answer is option 'A'. Can you explain this answer?.
The recurrence equationT(1) = 1T(n) = 2T(n - 1) + n, n ≥ 2evaluates toa)2n + 1- n - 2b)2n- nc)2n + 1- 2n - 2d)2n- nCorrect answer is option 'A'. Can you explain this answer? for Computer Science Engineering (CSE) 2025 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about The recurrence equationT(1) = 1T(n) = 2T(n - 1) + n, n ≥ 2evaluates toa)2n + 1- n - 2b)2n- nc)2n + 1- 2n - 2d)2n- nCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The recurrence equationT(1) = 1T(n) = 2T(n - 1) + n, n ≥ 2evaluates toa)2n + 1- n - 2b)2n- nc)2n + 1- 2n - 2d)2n- nCorrect answer is option 'A'. Can you explain this answer?.
Solutions for The recurrence equationT(1) = 1T(n) = 2T(n - 1) + n, n ≥ 2evaluates toa)2n + 1- n - 2b)2n- nc)2n + 1- 2n - 2d)2n- nCorrect answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for Computer Science Engineering (CSE).
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