is 1+z +z^2+z^3+z^4 1/(1-z)? I) z0 II) Z

Question

is 1+z +z^2+z^3+z^4> 1/(1-z)?

I) z>0
II) Z<1

Zoelef · Answer

z^4 + z^3 + z^2 + z + 1 > (1 / (1-z))? I) z > 0 II) z < 1 Let z = 0.5 1/16 + 1/8 + 1/4 + 1/2 + 1 = 31/16 1 / (1/2) = 2 31/16 < 2, nullifying both I and II.

believe2 · Answer

Is it C? the expression can be reduced to : 1+z +z^2+z^3+z^4 = z^5/(1-z) < 0 ? i. z > 0 Now, z^5/(1-z) < 0 can be true when z > 1 but false if z < 1 - insuff ii. Z < 1 the expression is true or false again depending on the value of z - insuff i & ii - sufficient

giddi77 · Answer

Nandan it can't be A since the right side if the inequality is undefined for z =1. Think again!

ywilfred · Answer

1) If z = 1/2, then 1+z+z^2+z^3+z^4 = 1 + 1/2 + 1/4 + 1/8 + 1/16 = 31/16 while 1/(1-z) = 2. So 1+z+z^2+z^3+z^4 < 1/(1-z). but if z = 2, then 1+z+z^2+z^3+z^4 = 1+2+4+8+16 = 31 while 1/(1-z) = -1 and 1+z+z^2+z^3+z^4 > 1/(1-z). Insufficient.

2) If z = 1/2, then we have 1+z+z^2+z^3+z^4 < 1/(1-z) but if z = 0, then LHS = RHS. Insufficient.

Using 1) and 2), we know z must lie between 0 and 1. It has to be a positive fraction and the inequality must be 1+z+z^2+z^3+z^4 < 1/(1-z). Sufficient.

Ans C

Gordon · Answer

Got C. I think the above inequality can be simplified to (1+z)*(1-z)>1 Things become clear once you reach this equation

razrulz · Answer

yep it is C we need both to find... the answer

joemama142000 · Answer

the oa is C

How did you guys simplify the equation? can you explain in detail? thanks in advance.

believe2 · Answer

I just used the following method to simplify: 1+z +z^2+z^3+z^4 > 1/(1-z) .......if you look at the left of the > sign, U'll notice that it is a geometric progression with first term 1 and the common ratio z. So you can use the formula for sum in a GP: The sum of the first n terms of a g.p. is Sn =a ( 1 - r^n)/(1-r) Hence, our inequality becomes: 1(1-z^5)/(1-z) < 1/(1-z) => (1 - z^5 -1)/(1-z) > 0 => - z^5/(1-z) > 0 => z^5/ (1-z) < 0 and so on.... Hope it helps!

joemama142000 · Answer

thanks believe

do you guys suggest learning all these geometric progression equations? I am noticing that in the more difficult problems, they come in handy. However, gmat doesnt stress the usage of them.

is 1+z +z^2+z^3+z^4> 1/(1-z)? I) z>0 II) Z<1

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