Maths question?

Question

1. The sum of the first and third terms of a geometric progression is 2/1/2.Given that the first term is 1/2,find the possible values of the common ratio,r, of the progression. In the case where r > 0,determine the last term of the progression which is less than 1 000

2. Given that p,q and r are three consecutive positive terms of a geometric progression. Express q in terms of p and r.

3. x,y,2/3,z and 2/27 are five consecutive terms of a geometric progression.Find the possible values of x,y and z.

4. The ratio of the sum of the second and third tems of geometric progression to its fourth term is 3:4. Show that 3r² - 4r - 4 = 0, where r is the common ratio. If r > 0 and the fifth term of this progression is 48, find the tenth term.

lol! i know this is a long question but please help me. Explain to me would be better. ^o^"'

thx to everyone who answered. (^_^)

lol! thx for the joke. (^_^)
but i still need the steps to do it. T^T

To Puggy:

that was 2 and a half. Was it confusing? (o.0)

Answer 1

1. Your question is worded confusing. What is 2/1/2?

2. If p, q, r and three consecutive positive terms of a GP, it follows that if we let t be the ratio,

q = pt
r = pt^2

Therefore, q/p = t and (r/p) = t^2, so

(r/p) = (q/p)^2
r/p = q^2 / p^2
(r/p)p^2 = q^2
rp = q^2, meaning q = sqrt(rp) {since q is positive}.

3.

x, y, 2/3, z, 2/27 are five consecutive terms of a GP.

Let r be the common ratio. Then, since z comes after 2/3,
z = (2/3)r. However, z also comes before 2/27. Therefore

(2/3)r^2 = 2/27. Multiplying both sides by 3/2, we get

r^2 = (2/27)(3/2)
r^2 = 1/9, so taking the square root of both sides,
r = {-1/3, 1/3}

This means we can get z;
z = (2/3) (-1/3) OR (2/3)(1/3)
z = {-2/9, 2/9}

We can also get y; since it is before 2/3, we have to DIVIDE by the ratio r. Dividing by the ratio (either -1/3 or 1/3) means multiplying by its reciprocal (either -3 or 3), so

y = (2/3)(-3) OR (2/3)(3)
y = {-2, 2}

To solve for x, we note that it preceeds TWO terms of (2/3), so we have to divide by r^2. Since r^2 = -1/3 or 1/3, r^2 = 1/9 regardless, so we divide by 1/9 (or multiply by 9).

x = (2/3)(9) = 9

To summarize: x = 9, y = {-2, 2}, z = {-2/9, 2/9}

Answer 2

1. 2/1/2 doesn't make sense, you'll have to reword it.

2. In a geometric progression, you multiply a constant number (say m) to get the next number. So, p * m = q, and q * m = r, or
q/p = m
r/q = m
q/p = r/q
q*q = r * p
q = sqrt(r*p)

3. y/x = m
2/3y = m
3z/2 = m
2/27z = m
3z/2 = 2/27z
3z*z = 4/27
z*z = 4/81
z = 2/9
m = 1/3
y = 2
x = 6
x = 6, y = 2, z = 2/9

4. Let x be the 2nd term, and r*x be the 3rd term. r*r*x is the 4th term.
(x + r*x) / r*r*x = 3/4
x(1 + r) / r*r*x = 3/4
(1 + r) / r*r = 3/4
4(1+r) = 4 + 4r= 3r*r
3r*r - 4r - 4 = 0
(3r - 2)(r + 2) = 0, r = -2 or 2/3
For r = 2/3, and fifth term is 48, the tenth term is 48 * (2/3)^5 = 32 * (2/3)^4 = 512/81 = 6 26/81

Answer 3

1)If I read 2/1/2 as 2/(1/2)=4then
the three terms are :1/2,1/2*r and 1/2*r^2.So
1/2+1/2*r^2 =4 .So 1/2*(1+r^2)=4 .So r is sqrt(7) if you take the positive root.
1/2*r^n mast bee less than 1000.So taking log
log1/2 +nlog r < log1000=3.
n=3.3010/log (sqrt 7)=7.8 So the las term lees than 1000 is the seventh.
2) if s is the ratio the three terms are p,p*s and p*s^2.So r/q=s
and r/p=s^2.So (r/q)^2=r/p r^2/q^2=r/p r*p=q^2 so q=sqrt(rp)
3) x ,x*r,x*r^2,x*r^·3 and x*r^4
x*r^2=2/3
x*r^4=2/27 .Dividing r^2=1/9 ===> r=+-1/3
x*1/9=2/3 x=6. y= 2,z=2/9, taking the+.You can make the same with -
4)( a*r+ar^2)/a*r^3 =3/4 You can simplify a*r and get (1+r)/r^2=3/4
so 3r^2-4r-4=0

Answer 4

if you need the answer to this question please call 1-800-tan (a/b)^2+cosec90- 1234+4321

:)

just a joke!