Consider the sequence defined a_0 = a_1 = a2 = 1, a_3 = 2. (a) Write out the fir
ID: 3123764 • Letter: C
Question
Consider the sequence defined a_0 = a_1 = a2 = 1, a_3 = 2. (a) Write out the first 12 terms (b) Find an explicit formula for the (c) Verify that a_7 and a_10 given by your computed in part (a). Solve the following recurrence relations with initial condition (a) a_n = a_n - 1 - a_n - 2 + a_n - 3, a_1 = i for i = 1, 2, 3. (b) a_n = a_n - 1 + a_n - 2 - a_n - 3, a_0 = a_1 = 1, a_2 = 2. Consider the sequence defined by a_n = a^1_n - 1/a^2_n - 2, with a_0 = 1 and a_1 = 2. (a) Write down the first five terms of the sequence. (b) Find an explicit formula for the terms of the sequence. Consid defined by a_n = (Squareroota_n - 1 + 2Squareroota_n - 2)^2, with a_0 =Explanation / Answer
24. (a) (The answer is clearly in ai = i)
an - an-1 + an-2 - an-3 = 0
Characteristic equation is
=> r3 - r2 + r - 1 = 0
=> r2 (r-1) + 1(r-1) = 0
=> (r2+1) (r-1) = 0
=> r2 = -1 or r = 1
=> r = 1
So the closed form is an = a + bn + cn2
Given a1 = 1
=> a + b*1 + c*12 = 1
=> a + b + c = 1 (1)
a2 = 2
=> a + b*2 + c*22 = 2
=> a + 2b + 4c = 2 (2)
Subtracting (1) from (2)
=> b + 3c = 1 (4)
a3 = 3
=> a + b*3 + c32 = 3
=> a + 3b + 9c = 3 (3)
Subtracting (1) from (3)
=> 2b + 8c = 2
=> b + 4c = 1 (4)
Subtracting (3) from (4)
=> c = 0
Substituting in (4)
=> b = 1
Substituting in (1)
a + 1 + 0 = 1
=> a = 0
So the closed form is 0 + 1n + 0n2
=> an = n
(b) (Clearly the terms are in AP)
an - an-1 - an-2 + an-3 = 0
Characteristic equation is
=> r3 - r2 - r + 1 = 0
=> r2 (r-1) - 1(r-1) = 0
=> (r2-1) (r-1) = 0
=> r2 = 1 or r = 1
=> r = 1 or r = -1 or r = 1
So the closed form is an = a1n + bn1n + c(-1)n
=> an = a + bn + c(-1)n
a0 = 1
=> a + b*0 + c(-1)0 = 1
=> a + c = 1 (1)
a1 = 1
=> a + b*1 + c(-1)1 = 1
=> a + b - c = 1 (2)
a2 = 2
=> a + b*2 + c(-1)2 = 2
=> a + 2b + c = 2 (3)
Substituting (1) in (3)
=> 1 + 2b = 2
=> 2b = 1
=> b = 1/2
Substituting in (1)
=> a + 1/2 -c = 1
=> a - c = 1/2 (4)
Adding (1) and (4)
=> 2a = 3/2
=> a = 3/4
Substituting in (1)
=> 3/4 + c = 1
=> c = 1/4
So the solution is an = 3/4 + n/2 +1/4(-1)n