SOLUTION: Pls help me solve this: 1): A G.P is such that the 3rd term minus the first term is 48.the 4th minus the second term is 44.find the common ratio, the first term and the 12th term.

Algebra ->  Formulas -> SOLUTION: Pls help me solve this: 1): A G.P is such that the 3rd term minus the first term is 48.the 4th minus the second term is 44.find the common ratio, the first term and the 12th term.      Log On


   

Question 1139299: Pls help me solve this:
1): A G.P is such that the 3rd term minus the first term is 48.the 4th minus the second term is 44.find the common ratio, the first term and the 12th term.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
A G.P is such that the 3rd term minus the first term is 48.the 4th minus the second term is 44.find the common ratio, the first term and the 12th term.
A G.P:+a%5B1%5D, a%5B1%5D%2Ar , a%5B1%5D%2Ar%5E2, a%5B1%5D%2Ar%5E3,......., a%5B1%5D%2Ar%28n-1%29

the 3rd term minus the first term is 48:

a%5B3%5D-a%5B1%5D=48......express 3rd term in terms of first term

a%5B1%5Dr%5E2-a%5B1%5D=48

a%5B1%5D%28r%5E2-1%29=48

a%5B1%5D=48%2F%28r%5E2-1%29.......eq.1

the 4th minus the second term is 44:
a%5B4%5D-a%5B2%5D=44.....express 4th and 2nd term in terms of first term

a%5B1%5D%2Ar%5E3-a%5B1%5D%2Ar=44

a%5B1%5D%28r%5E3-r%29=44

a%5B1%5D=44%2F%28r%5E3-r%29

a%5B1%5D=44%2Fr%28r%5E2-1%29.......eq.2

from eq.1 and eq.2 we have

48%2F%28r%5E2-1%29=44%2Fr%28r%5E2-1%29.....simplify

48%2Fcross%28%28r%5E2-1%29%29=44%2Fr%28cross%28%28r%5E2-1%29%29%29

48%2F1=44%2Fr.......cross multiply

48r=44

r=44%2F48

r=11%2F12-> common ratio


go to a%5B1%5D=48%2F%28r%5E2-1%29.......eq.1, substitute r

a%5B1%5D=48%2F%28%2811%2F12%29%5E2-1%29

a%5B1%5D=48%2F%28121%2F144-1%29

a%5B1%5D=48%2F%28121%2F144-144%2F144%29

a%5B1%5D=48%2F%28-23%2F144%29

a%5B1%5D=-6912%2F23

so,

a%5B1%5D=-6912%2F23

a%5B2%5D=-%286912%2F23%29%2811%2F12%29=>a%5B2%5D=-6336%2F23

a%5B3%5D=-%286912%2F23%29%2811%2F12%29%5E2=>a%5B2%5D=-5808%2F23

a%5B4%5D=-%286912%2F23%29%2811%2F12%29%5E3=>a%5B2%5D=-5324%2F23

check differences:

a%5B3%5D-a%5B1%5D=48=> -5808%2F23-%28-6912%2F23%29=48->true
a%5B4%5D-a%5B2%5D=44=>-5324%2F23-%28-6336%2F23%29=44->true

now use general formula to find 12th term:

a%5Bn%5D=a%5B1%5D%2Ar%5E%28n-1%29.......plug in known values

a%5B12%5D=-%286912%2F23%29%2A%2811%2F12%29%5E%2812-1%29

a%5B12%5D=-%286912%2F23%29%2A%2811%2F12%29%5E11

a%5B12%5D=-%286912%2F23%29%2A%28285311670611%2F743008370688%29

a%5B12%5D=-285311670611%2F2472394752