I'm partial to solving with generating functions. Let
Multiply both sides of the recurrence by and sum over all .
Shift the indices and factor out powers of as needed so that each series starts at the same index and power of .
Now we can write each series in terms of the generating function . Pull out the first few terms so that each series starts at the same index .
Solve for :
Splitting into partial fractions gives
which we can write as geometric series,
which tells us
# # #
Just to illustrate another method you could consider, you can write the second recurrence in matrix form as
By substitution, you can show that
or
Then solving the recurrence is a matter of diagonalizing the coefficient matrix, raising to the power of , then multiplying by the column vector containing the initial values. The solution itself would be the entry in the first row of the resulting matrix.
Answer:
x = -7
Step-by-step explanation:
3x + 13 = -2x - 22
subtract 13 from both sides
3x = -2x - 35
Add 2x to both sides
5x = -35
Divide each side by 5
x = -7
Jasmine is running faster.
Since the track is 1/5 of a km and she can run 1km in 7 minutes, divide seven to get 1.4.
This means that Jasmine can run one lap in 1.4 minutes.
Since a Kyah can run 6 laps in 10 minutes, do 10 divided by 6 and the answe will be 1.6 repeating.
1.4 < 1.6 so Jasmine is faster
33 is fifteen less than k
---"Is" can be a key word in this statement, meaning equals
33 = k - 15
k = 48
Hope this helps!