The sum of the first three terms of G.P. is 7 and sum of their squares is 21. Calculate first five terms of the G.P.

For the beginning, we'll note the terms of the G.P.
as:

a, a*r, a*r^2, ...

From
enunciation, we know that the sum of the first 3 terms
is:

a + a*r + a*r^2 = 7

We'll
factorize and we'll get:

a(1 + r + r^2 ) = 7
(1)

Also, the sum of the squares of the first 3 terms
is:

a^2 + (a*r)^2 + (a*r^2)^2 =
21

We'll factorize and we'll
get:

a^2(1 + r^2 + r^4) = 21
(2)

We'll form the ratio
(2)/(1)^2:

a^2(1 + r^2 + r^4)/[a(1 + r + r^2 )]^2 =
21/7^2

We'll eliminate like
terms:

(1 + r^2 + r^4)/(1 + r + r^2 )^2 =
3/7

We'll form the square at
numerator:

1 + r^2 + r^4 = (1 + 2r^2 + r^4) - r^2 =
(1+r^2)^2 - r^2

Now, the result is a difference of
squares:

(1+r^2)^2 - r^2 =
(1+r^2+r)(1+r^2-r)

With this result, we'll go back into the
ratio (2)/(1)^2:

(1+r^2+r)(1+r^2-r) / (1 + r + r^2 )^2 =
3/7

We'll eliminate like
brackets:

(1+r^2-r) / (1 + r + r^2 ) =
3/7

We'll cross multiply:

7 +
7r^2 - 7r = 3 + 3r + 3r^2

We'll move all terms to the left
side:

7 + 7r^2 - 7r - 3 - 3r - 3r^2 =
0

We'll combine like
terms:

4r^2 - 10r + 4 =
0

We'll divide by 2 :

2r^2 -
5r + 2 = 0

We'll apply the quadratic
formula:

r1 =
[5+sqrt(25-16)]/4

r1 =
(5+3)/4

r1 = 2

r2 =
(5-3)/4

r2 = 1/2

For r = 2,
we'll calculate the first term of the g.p. from the relation
(1):

a(1 + r + r^2 ) = 7

a(1 +
2 + 4 ) = 7

7a = 7

a =
1

So, the g.p. is:

1 , 1*2 ,
1*2*2 , ..........

For r = 1/2, the first term
is:

a(1 + r + r^2 ) = 7

a(1 +
1/2 + 1/4 ) = 7

a*(7/4) = 7

a
= 4

The g.p. is:

4 , 4*(1/2),
4*(1/2)^2,..................