In a sequence of four positive numbers, the first three are in geometric progression and the last three are in arithmetic progression. The first number is 12 and the last number is 452. The sum of the two middle numbers can be written as ab where a and b are coprime positive integers. Find a+b.

Question

cwrw238 · Answer

the first 3 numbers  are
12 12r and 12r^2    where r = common ratio of the G P

last 3 are 12r , 12r + d and 12r + 2d   where d = common difference of the A P

12r + 12d = 45

cwrw238 · Answer

* 12r + 2d = 452

amistre64 · Answer

my idea:

12, 12+k, 12+2k

(12+k)r = 12+2k

12, 12+k, 12+2k, (12+k)r^2 = 452

(12+k)r^2 = 452

amistre64 · Answer

... lol, i got that backwards dont i

cwrw238 · Answer

yea lol

cwrw238 · Answer

it comes down to solving  system of 2 equations, i think

12r + 2d = 452
and  12r + d = 12r^2  ( the third term)

anonymous · Answer

12, 12r, 12r^2, 452.
since the last three terms 12r, 12r^2 and 452 are in arithmetic progression,
12r^2=(12r+452)/2

cwrw238 · Answer

from first equation 2d = 452 - 12r
d = 226 - r
plug this into second equation to get
12r + 226 - r = 12r^2

12r^2 - 11r - 226 = 0

cwrw238 · Answer

ooohh - i see i have have made a mistake
d = 226 - 6r

anonymous · Answer

@cwrw238  yea..right.
even i'm getting the same thing! :)

cwrw238 · Answer

lol

cwrw238 · Answer

common ratio r satisfies the equation

12r^2 - 6r - 226 = 0

cwrw238 · Answer

solving this gives r = 4.569  or  -4.0969

cwrw238 · Answer

we can ignore the negative root because the 4 numbers are all postive

d = (452 - 12(4.5969) ) / 2 = 198.4186

this makes the sequence

12, 55.1628, 253.5814, 452