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A sequence is defined by the recurrence relation an-(p+q)an-1-(pq)an-2, where p and q-5. The first two terms of the sequence are a1=-1 and a2-2.

A sequence is defined by the recurrence relation an-(p+q)an-1-(pq)an-2, where p and q-5. The first two terms of the sequence are a1=-1 and a2-2.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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sequences, that is: an-c1r-1+c2r2-1, where c1 and c2 still need to be determined - to determine the constants c1 and c2, use the starting values a and a2given above to set up a system of 2 equations for the 2 values c1 and c2 - finally, solve the system of 2 equations to determine c1 and c2 Put all the parts together and you should now have a closed formula for the sequence in the form a,=cr,n-1+cr2n-1, where the growth rates r and r2 and the constants C and c2 are all known values. C2 To verify that you have correctly solved this recurrence relation and constructed the correct closed formula, in the box below enter the ratio of the 2 constants cl (If your answer is a decimal, round it correctly to 4 decimal places.) Answer:
Transcribed Image Text:sequences, that is: an-c1r-1+c2r2-1, where c1 and c2 still need to be determined - to determine the constants c1 and c2, use the starting values a and a2given above to set up a system of 2 equations for the 2 values c1 and c2 - finally, solve the system of 2 equations to determine c1 and c2 Put all the parts together and you should now have a closed formula for the sequence in the form a,=cr,n-1+cr2n-1, where the growth rates r and r2 and the constants C and c2 are all known values. C2 To verify that you have correctly solved this recurrence relation and constructed the correct closed formula, in the box below enter the ratio of the 2 constants cl (If your answer is a decimal, round it correctly to 4 decimal places.) Answer:
A sequence is defined by the recurrence relation an-(p+q)an-1-(pq)an-2, where p-D4 and q=5. The first two terms of the sequence are a1=-1 and a2=2. To determine a closed formula for the terms of the sequence, we can follow the procedure we discussed in class: • substitute a geometric sequence an=rn-1 into the recurrence relation (and also substitute appropriately adjusted expressions for an-1 and an-2) • collect all the terms onto one side of the equation and factor out a common factor of rn-3 • solve the resulting quadratic equation to get 2 possible geometric growth rates, r, and r2 • our closed formula must therefore be be some combination of 2 geometric sequences, that is: a,=c1r1"-1+c2r2 c1 and c2 still need to be n-1 where
Transcribed Image Text:A sequence is defined by the recurrence relation an-(p+q)an-1-(pq)an-2, where p-D4 and q=5. The first two terms of the sequence are a1=-1 and a2=2. To determine a closed formula for the terms of the sequence, we can follow the procedure we discussed in class: • substitute a geometric sequence an=rn-1 into the recurrence relation (and also substitute appropriately adjusted expressions for an-1 and an-2) • collect all the terms onto one side of the equation and factor out a common factor of rn-3 • solve the resulting quadratic equation to get 2 possible geometric growth rates, r, and r2 • our closed formula must therefore be be some combination of 2 geometric sequences, that is: a,=c1r1"-1+c2r2 c1 and c2 still need to be n-1 where
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