Discrete Mathematics: Please Help me with Question 3 (See attachment) 1. Compute the followingа) П-1)*b) 2+5+8+11+..+302+305ic) E100Li=11000!d)998!2. for the following recursive functions, find f(1), f(2), f(3), and f(4)f(0) = 2f(k) =k-(f(k-1))²F(1) = 1- (((0))² = 1 – 4 = -3F(2) = 2- ((1))? = 2 - (-3) = 5F(3) = 3 – ((2))² = 3 – 25 = -22F(4) = 4 - ((4))² = 4 – (-22)² = 4 – 484 = -400n' +n3. Prove that S,is a solution of the recursive relation2S, = 1S = S1 +k for k>1k-14. Using pattern recognition, solve the recursive relationf(0) = 4f(k) = 5+1.1f(k– 1) for k>0b) Evaluate f(10)5. Prove: n' + 5nis divisible by 6 for all integer n20.6. Use mathematical induction to prove that2+5+8+11+......+(3n-1)-n(3n+1)/27. A person borrowed $4000 on a bank credit card at a nominal rate of 24% per year,whichis actually charged at a rate of 2% per month.a) what is the effective annual percentage rate (Effective APR) for the card?b) Assume that the person does not place any additional charges on the card andpays the bank $300 each month to pay off the loan. Let B(n) be the balance owedon the card after n months. Find explicit formula for B(n).c) How long will be required to pay off the debt?d) What is the total amount of money the person will have paid for the loan?8. Solve the recursion:A = 1; A2 = -1Ar = 5Ak-1-6Ax-2x+19. Let f(x) =%3!2x-4a) Find domain of f(x)b) Prove that f(x) is one-to-one

Answered: n? +n 3. Prove that S, = is a solution…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Discrete Mathematics: Please Help me with Question 3 (See attachment)

1. Compute the following
а) П-1)*
b) 2+5+8+11+.
.+302+305
i
c) E
100
Li=1
1000!
d)
998!
2. for the following recursive functions, find f(1), f(2), f(3), and f(4)
f(0) = 2
f(k) =k-(f(k-1))²
F(1) = 1- (((0))² = 1 – 4 = -3
F(2) = 2- ((1))? = 2 - (-3) = 5
F(3) = 3 – ((2))² = 3 – 25 = -22
F(4) = 4 - ((4))² = 4 – (-22)² = 4 – 484 = -400
n' +n
3. Prove that S,
is a solution of the recursive relation
2
S, = 1
S = S1 +k for k>1
k-1
4. Using pattern recognition, solve the recursive relation
f(0) = 4
f(k) = 5+1.1f(k– 1) for k>0
b) Evaluate f(10)
5. Prove: n' + 5nis divisible by 6 for all integer n20.
6. Use mathematical induction to prove that
2+5+8+11+......+(3n-1)-n(3n+1)/2
7. A person borrowed $4000 on a bank credit card at a nominal rate of 24% per year,
which
is actually charged at a rate of 2% per month.
a) what is the effective annual percentage rate (Effective APR) for the card?
b) Assume that the person does not place any additional charges on the card and
pays the bank $300 each month to pay off the loan. Let B(n) be the balance owed
on the card after n months. Find explicit formula for B(n).
c) How long will be required to pay off the debt?
d) What is the total amount of money the person will have paid for the loan?
8. Solve the recursion:
A = 1; A2 = -1
Ar = 5Ak-1-6Ax-2
x+1
9. Let f(x) =
%3!
2x-4
a) Find domain of f(x)
b) Prove that f(x) is one-to-one

Expert Solution

Step 1

3. Given,

$S_{1} = 1, S_{k} = S_{k - 1} + k; for k > 1$

Step 2

Substitute $k = 2, 3, 4, \dots$ in the given recurrence relation,

$S_{2} = S_{1} + 2 = 1 + 2 = 3 S_{3} = S_{2} + 3 = 3 + 3 = 6 S_{4} = S_{3} + 4 = 6 + 4 = 10 \dots \dots \dots \dots \dots \dots \dots \dots$

Now

$1 = \frac{1^{2} + 1}{2}, 3 = \frac{2^{2} + 2}{2}, 6 = \frac{3^{2} + 3}{2}, 10 = \frac{4^{2} + 4}{2}, \dots \dots \dots \dots \dots \dots \dots \dots$

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