# Thread: Arithmetic and Geometric Progressions

1. ## Arithmetic and Geometric Progressions

Positive numbers a1, a2, a3 are in arithmetic progression, while positive numbers g1, g2, g3 are in geometric progression. Given that a1 + g1 = 26, a2 + g2 = 50, a3 + g3 = 114, and a1 + a2 + a3 = 60, find both progressions.

***Please note the 1, 2 and 3 are meant to be subscripts

Arithmetic Progression: a2 - a1 = a3 - a2
Geometric Progression: g2/g1 = g3/g2

Just a few hints. There are more.

3. let, a1= a; a2 = a+d; a3 = a+2d and g1 = x; g2= x r; g3 = x r^2
d: different and r: rasio.
substitute the equality to the problem.
ex:
a1 + g1 = a + x = 36... (1)
.
.
.
etc

hope it'll help ^_^

4. Originally Posted by chris520
a1 + g1 = 26, a2 + g2 = 50, a3 + g3 = 114, and a1 + a2 + a3 = 60.....
From these, can you "see" that g1 + g2 + g3 = 130?

5. Originally Posted by chris520
Positive numbers a1, a2, a3 are in arithmetic progression, while positive numbers g1, g2, g3 are in geometric progression. Given that a1 + g1 = 26, a2 + g2 = 50, a3 + g3 = 114, and a1 + a2 + a3 = 60, find both progressions.

***Please note the 1, 2 and 3 are meant to be subscripts
Let the Arithmetic progression be
a2-d, a2, a2+d
so that (3)a2=60 so a2=20.

Then g2=30.

Let the Geometric progression be
(g2)/r, g2, r(g2)

Then a1+a2+a3+g1+g2+g3=190=60+130
so g1+g2+g3=130

(g2)/r+g2+(r)g2=130
30+30r+30r^2=130r
30r^2-100r+30=0
3r^2-10r+3=0
(3r-1)(r-3)=0

allows you to solve for "r" and write out the Geometric sequence.

You can calculate the terms of the Arithmetic sequence from those.