Question

4.

Let an be the sequence defined by a0 = 0 and an = 2an−1 + 2 for n > 1.

(a) Find the value of sum 4 i=0 ai .

(b) Use induction to prove that an = 2n+1 − 2 for all n ∈ N.

Answer #1

Suppose that a sequence an (n = 0,1,2,...) is defined
recursively by a0 = 1, a1 = 7, an = 4an−1 − 4an−2 (n ≥ 2). Prove by
induction that an = (5n + 2)2n−1 for all n ≥ 0.

Let a0 = 1, a1 = 2, a2 = 4, and an = an-1 + an-3 for n>=
3.
Let P(n) denote an an <= 2^n.
Prove that P(n) for n>= 0 using strong induction:
(a) (1 point) Show that P(0), P(1), and P(2) are true, which
completes
the base case.
(b) Inductive Step:
i. (1 point) What is your inductive hypothesis?
ii. (1 point) What are you trying to prove?
iii. (2 points) Complete the proof:

Let a0 = 1, a1 = 2, a2 = 4, and an = an-1 + an-3 for n>=
3.
Let P(n) denote an an <= 2^n.
Prove that P(n) for n>= 0 using strong induction:
(a) (1 point) Show that P(0), P(1), and P(2) are true, which
completes
the base case.
(b) Inductive Step:
i. (1 point) What is your inductive hypothesis?
ii. (1 point) What are you trying to prove?
iii. (2 points) Complete the proof:

. Consider the sequence defined recursively as a0 = 5, a1 = 16
and ak = 7ak−1 − 10ak−2 for all integers k ≥ 2. Prove that an = 3 ·
2 n + 2 · 5 n for each integer n ≥ 0

Let A = (A1, A2, A3,.....Ai) be defined as a sequence containing
positive and negative integer numbers.
A substring is defined as (An, An+1,.....Am) where 1 <= n
< m <= i.
Now, the weight of the substring is the sum of all its
elements.
Showing your algorithms and proper working:
1) Does there exist a substring with no weight or zero
weight?
2) Please list the substring which contains the maximum weight
found in the sequence.

1) Find the sum S of the series where S = Σ i ai -- here i
varies from 1 to n.
Use the mathematical induction to prove the following:
2) 13 + 33 + 53 + …. + (2n-1)3 = n2(2n2-1)
3) Show that n! > 2n for all n > 3.
4) Show that 9(9n -1) – 8n is divisible by 64.
Show all the steps and calculations for each of the above
and explain your answer in...

) Let α be a fixed positive real number, α > 0. For a
sequence {xn}, let x1 > √ α, and define x2, x3, x4, · · · by the
following recurrence relation xn+1 = 1 2 xn + α xn (a) Prove that
{xn} decreases monotonically (in other words, xn+1 − xn ≤ 0 for all
n). (b) Prove that {xn} is bounded from below. (Hint: use proof by
induction to show xn > √ α for all...

Consider the following sequence: 0, 6, 9, 9, 15, 24, . . .. Let
the first term of the sequence, a1 = 0, and the second, a2 = 6, and
the third a3 = 9. Once we have defined those, we can define the
rest of the sequence recursively. Namely, the n-th term is the sum
of the previous term in the sequence and the term in the sequence 3
before it: an = an−1 + an−3. Show using induction...

Solve the recurrence relation:
an = 3an−1 − 2an−2 + 3n
with a0 = 1, a1 = 0.

solve the non-homogenous recurrence relation for
an =
2an-1+an-2-2an-3+8.3n-3 where
a0 = 2, a1 = 6 ve
a2=13
Find characteric equation by plugging
in an = rn
try to solve general solution and solve nonhomogeneous
particular solution
and find total final answer please..
My book anwer is
A(1)n+B(-1)n+C(2)n+k3n
, A=1/2, B=-1/2, C=1 ve k=1.
can you give me more explain about this please..?

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