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Can anyone pls help me to solve question 2 f and g and pls provide me a explanation I’m with that questions for three days

Can anyone pls help me to solve question 2 f and g and pls provide me a explanation-example-1
User Cdn
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Answer:

f) a[n] = -(-2)^n +2^n

g) a[n] = (1/2)((-2)^-n +2^-n)

Explanation:

Both of these problems are solved in the same way. The characteristic equation comes from ...

a[n] -k²·a[n-2] = 0

Using a[n] = r^n, we have ...

r^n -k²r^(n-2) = 0

r^(n-2)(r² -k²) = 0

r² -k² = 0

r = ±k

a[n] = p·(-k)^n +q·k^n . . . . . . for some constants p and q

We find p and q from the initial conditions.

__

f) k² = 4, so k = 2.

a[0] = 0 = p + q

a[1] = 4 = -2p +2q

Dividing the second equation by 2 and adding the first, we have ...

2 = 2q

q = 1

p = -1

The solution is a[n] = -(-2)^n +2^n.

__

g) k² = 1/4, so k = 1/2.

a[0] = 1 = p + q

a[1] = 0 = -p/2 +q/2

Multiplying the first equation by 1/2 and adding the second, we get ...

1/2 = q

p = 1 -q = 1/2

Using k = 2^-1, we can write the solution as follows.

The solution is a[n] = (1/2)((-2)^-n +2^-n).

User Thlgood
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