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1st week of my university and i got the painful assignment. I've not touched math from last 4-5 months, so i'm having trouble with few questions. Please help me out, thanks in advance

1- Prove that if x not equal 0, then [tex] x + 2x^{2} + 3x^{3} + ... + nx^{n}(1+x) = [ x - (n - 1)x^{n+1} + nx^{n+2} ] / (1-x)^{2} [/tex] for every poitive integer n.

2-Prove by induction that if [tex]x \geq 0[/tex] [tex](1+x) ^n \geq 1+nx[/tex]

3- Using the fact that [tex]d/dx(x) = 1 [/tex] and the product rule, prove by induction that [tex]d/dx(x^{n}) = nx^{n-1}[/tex] for every poitive integer n.

4- (a) Treating the equation 4x^2 +4xy +2y^2 − 4x − 2y +1 = 0 as a quadratic equation in x with coefficients in terms of y, solve for x.

b) Disprove the statement “For all real numbers x and y, 4x^2 +4xy +2y^2 − 4x − 2y +1 = 0”.

1st question: i tried n = 1 & then n = k but it didn't work out

2nd question: here's what i've

n = 0 (1 + x)^0 ≥ 1 + 0x. And that's true!

assume the expression is true when n = k

(1 + x)^k ≥ 1 + kx.

Prove for n = k + 1

(1 + x)^k + 1 ≥ 1 + (k + 1)x.

So (1 + x)^k + 1 = (1 + x)^k (1 + x)

From the 1st hypothesis

Therefore, (1 + x)^k (1 + x) ≥ (1 + kx) (1 + x)

after this i've no idea how to solve the rest of it

3rd question: no clue at all how to get started

4th question part a: since we need to put the who expression into ax^2 + bx + c so this is what i did

4x^2 + 4(y-1)x + 2y^2 - 2y + 1 = 0

therefore a = 4, b = 4(y-1) and c = 2y^2 - 2y + 1. I tried to use the quadratic formula but it doesn't give me a real root. Maybe i did something wrong and for part b i don't know how to solve it.

Please guys help me out as soon as possible, these question are due withing few days. All the help would be really apperciated , again tons of thanks in advance!

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# Prove the expressions by induction method

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