Description

Attempt all questions and show all your work. Attach to Honesty Declaration Form.

1. use the mathematical induction on integer n to prove each of the following;

(a)  1(4) + 2(5) + 3(6) + … + n(n + 3) = ⅟3(n)(n + 1)(n + 5) for n ≥ 1 ;

(b)  3ⁿ⁺¹(n + 2)! ≥ 2ⁿ(n + 3)! for n ≥ 0 ;

(c)   (1 – ⅟3²)(1 – ⅟4²)… (1 – ) = ) for n ≥ 2 ;

(d)   2ᶟⁿ⁺² + 3⁶ⁿ⁺¹ is divisible by 7 for   n ≥ 1 .

2. Simplify as much as possible using properties of sigma notation.
3. identities

are given. Use the identities to evaluate the sum

4. Find solution of the following equation. Express your answers in polar form.

(⁴ + 6² + 9)(ᶟ + 5² + 4) = 0

Hint: In the right bracket consider 5²  as  ² + 4² and then solve it by factoring.

5. Express each of the following in simplified cartesian form.

(a)  (  –  )¹⁰ ;

(b)   ()¹¹ (1 – )⁸  (– 3)⁸ .

6. Find all solutions of the equation

. Express all solutions in polar form, simplified as much as possible.