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What is the recurrence relation for 1, 7, 31, 127, 499?
- a)bn+1 = 5bn-1+3
- b)bn = 4bn+7!
- c)bn = 4bn-1+3
- d)bn = bn-1+1
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
What is the recurrence relation for 1, 7, 31, 127, 499?a)bn+1= 5bn-1+3...
Look at the differences between terms: 1, 7, 31, 124,…. and these are growing by a factor of 4. So, 1⋅4 = 4, 7⋅4 = 28, 31⋅4 = 124, and so on. Note that we always end up with 3 less than the next term. So, bn = 4bn-1 + 3 is the recurrence relation and the initial condition is b0 = 1.
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What is the recurrence relation for 1, 7, 31, 127, 499?a)bn+1= 5bn-1+3...
Recurrence Relation for the given sequence:
The given sequence is 1, 7, 31, 127, 499. To find the recurrence relation, we need to identify the pattern between consecutive terms.
Let's observe the difference between consecutive terms:
7 - 1 = 6
31 - 7 = 24
127 - 31 = 96
499 - 127 = 372
We can notice that the differences between consecutive terms are increasing by a factor of 4.
Let's calculate the second difference:
24 - 6 = 18
96 - 24 = 72
372 - 96 = 276
The second differences are not constant, which indicates that the given sequence is not a quadratic sequence.
Based on the first differences, we can conclude that the given sequence is likely to be a cubic sequence.
Let's assume the nth term of the given sequence is given by bn.
Recurrence relation:
bn = bn-1 + (4n^2 - 4n + 1)
Explanation:
We can observe that the nth difference between consecutive terms is given by (4n^2 - 4n + 1).
So, to find the nth term, we can add the nth difference to the (n-1)th term.
For example, to find the 2nd term (7), we add the 2nd difference (6) to the 1st term (1):
7 = 1 + 6
Similarly, to find the 3rd term (31), we add the 3rd difference (24) to the 2nd term (7):
31 = 7 + 24
This pattern continues for the remaining terms.
Hence, the correct recurrence relation for the given sequence is:
bn = bn-1 + (4n^2 - 4n + 1)
The given sequence is 1, 7, 31, 127, 499. To find the recurrence relation, we need to identify the pattern between consecutive terms.
Let's observe the difference between consecutive terms:
7 - 1 = 6
31 - 7 = 24
127 - 31 = 96
499 - 127 = 372
We can notice that the differences between consecutive terms are increasing by a factor of 4.
Let's calculate the second difference:
24 - 6 = 18
96 - 24 = 72
372 - 96 = 276
The second differences are not constant, which indicates that the given sequence is not a quadratic sequence.
Based on the first differences, we can conclude that the given sequence is likely to be a cubic sequence.
Let's assume the nth term of the given sequence is given by bn.
Recurrence relation:
bn = bn-1 + (4n^2 - 4n + 1)
Explanation:
We can observe that the nth difference between consecutive terms is given by (4n^2 - 4n + 1).
So, to find the nth term, we can add the nth difference to the (n-1)th term.
For example, to find the 2nd term (7), we add the 2nd difference (6) to the 1st term (1):
7 = 1 + 6
Similarly, to find the 3rd term (31), we add the 3rd difference (24) to the 2nd term (7):
31 = 7 + 24
This pattern continues for the remaining terms.
Hence, the correct recurrence relation for the given sequence is:
bn = bn-1 + (4n^2 - 4n + 1)
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What is the recurrence relation for 1, 7, 31, 127, 499?a)bn+1= 5bn-1+3b)bn= 4bn+7!c)bn= 4bn-1+3d)bn= bn-1+1Correct answer is option 'C'. Can you explain this answer?
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