A roll of ribbon was 12 meters long. Diego cut 9 pieces of ribbon that were 0.4 meters each to tie some presents. He then used t
2 answers:
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1. Geometric Sequence
2.
3.
Step-by-step explanation:
Given sequence is:
6, 18, 54, 162,....
Here
(a) Is this an arithmetic or geometric sequence?
We can see that the difference between the terms is not same so it cannot be an arithmetic sequence.
We have to check for common ratio (ratio between consecutive terms of a sequence) denoted by r
As the common ratio is same, the given sequence is a geometric sequence.
(b) How can you find the next number in the sequence?
Recursive formulas are used to find the next number in sequence using previous term
Recursive formula for a geometric sequence is given by:
In case of given sequence,
So to find the 5th term
(c) Give the rule you would use to find the 20th week.
In order to find the pushups for 20th week, explicit formul for sequence will be used.
The general form of explicit formula is given by:
Putting the values of a_1 and r
Hence,
1. Geometric Sequence
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Keywords: Geometric sequence, common ratio
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First look for the fundamental solutions by solving the homogeneous version of the ODE:
The characteristic equation is
with roots and
, giving the two solutions
and
.
For the non-homogeneous version, you can exploit the superposition principle and consider one term from the right side at a time.
Assume the ansatz solution,
(You could include a constant term <em>f</em> here, but it would get absorbed by the first solution anyway.)
Substitute these into the ODE:
is already accounted for, so assume an ansatz of the form
Substitute into the ODE:
Assume an ansatz solution
Substitute into the ODE:
So, the general solution of the original ODE is