how do you solve this question:....?

Question

Find the 1st 5 terms of a Geometric series, the sum of the 1st and 3rd term is 50 and the sum of the 2nd and 4th term is 150. Please show your calculations. Thanks ^ ^

Answer 1

Let r be the common ratio between each term of the geometric sequence.

Let a be the first term in the sequence.

Then your sequence looks like: a, ar, ar^2, ar^3, ar^4, ...

We need to determine "a" and "r":

The sum of the first and third terms is 50:
50 = a + ar^2 = a(1+r^2)

The sum of the second and fourth terms is 150:
150 = ar + ar^3 = ar(1+r^2)

From the first equation, we see that 1+r^2 = 50/a
Substituting into the second equation, we get that 150 = ar(50/a) = 50r.
Thus, r = 3.

Plugging this back in to 1+r^2 = 50/a, we get 10 = 50/a.
This means that a = 5.

So, our sequence is a, ar, ar^2, ar^3, ar^4 with a = 5 and r = 3.
This translates the first five terms to: 5, 15, 45, 135, 405

Answer 2

Let r be the common ratio between each term of the geometric sequence.

Let a be the first term in the sequence.

Then your sequence looks like: a, AR, AR^2, AR^3, AR^4, ...

We need to determine "a" and "r":

The sum of the first and third terms is 50:
50 = a + AR^2 = a(1+r^2)

The sum of the second and fourth terms is 150:
150 = AR + AR^3 = AR(1+r^2)

From the first equation, we see that 1+r^2 = 50/a
Substituting into the second equation, we get that 150 = AR(50/a) = 50r.
Thus, r = 3.

Plugging this back in to 1+r^2 = 50/a, we get 10 = 50/a.
This means that a = 5.

So, our sequence is a, AR, AR^2, AR^3, AR^4 with a = 5 and r = 3.
This translates the first five terms to: 5, 15, 45, 135, 405

Answer 3

term 1 = a
term 2 = ar
term 3 = ar^2
term 4 = ar^3
term 5 = ar^4

term 1 + term 3 = a + ar^2 = 50
term 2 + term 4 = ar + ar^3 = 150

a(1 + r^2) = 50
ar(1+ r^2) = 150

a = 50 / (1 + r^2)
a = 150 /r(1 + r^2)

So 50/(1 + r^2) = 150/ r(1+ r^2)

rearranging
150 / 50 = r(1 + r^2)/(1 + r^2)
3 = r NB the 1+r^2 cancel out top and bottom.
So when r = 3 and substituting back into first equation.
We have a(1 + 3^2) = 50
a x 10 = 50
a = 5

So first five terms are
term 1 = 5 = 5
term 2 = 5 x 3 = 15
term 3 = 5 x 3^2 = 45
term 4 = 5 x 3^3 = 135
term 5 = 5 x 3^4 = 405