## Description

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

- 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 .

- Simplify as much as possible using properties of sigma notation.
- identities

are given. Use the identities to evaluate the sum

- 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.

- Express each of the following in simplified cartesian form.

(a) ( – )¹⁰ ;

(b) ()¹¹ (1 – )⁸ (– 3)⁸ .

- Find all solutions of the equation

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

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