GPs (1 Viewer)

[moo_moo_molly]

What number must be added to 1/10, 1/4 and 9/20 to make a GP...

I made 3 equations a=1/10+n ar=1/4+n and ar^2=9/20+n then simultaneously solved but the number i got was -1/10 which doesnt make sense because a would be equal to 0.

 

[webby234]

a = 1/10 + n
ar = 1/4 + n
ar^2 = 9/20 + n

n = a - (1/10)

ar - a = 3/20
r = (3/20 + a)/a

a[(3/20 + a)/a]^2 = 9/20 + a - 1/10

9/400 + 3a/10 + a^2 = 9a/20 + a^2 - a/10

9 + 120a = 180a - 40a

20a = 9

a = 9/20
n = 7/20
r = 4/3

So GP is 9/20, 12/20, 16/20
= 9/20, 3/5, 4/5

7/20 must be added to each.

There must be an easier way though.

 

[Riviet]

An alternate and possibly more faster way:
The series of the GP is 1/10 + n, 1/4 + n, 9/20 + n.
Now using the fact that T_n/T_n-1 = the common ratio r,
(9/20 + n) / (1/4 + n) = (1/4 + n) / (1/10 + n)
Cross multiply the equation and solve for n.
You should get n=7/20, add that to each of your terms from the original series and you have your GP.