I need help solving a geometric sequence which is defined by:
first term + third term = 280
fourth term + sixth term = 35
I need to find the ratio of the sequence and the first term, this problem is solved in the book in which it appears, but the solution says only:
q = ratio, a = first term
\(\displaystyle \large a + aq^2 = 280\\
aq^3 + aq^5 = 35\)
and thus
\(\displaystyle \large q^3 = \frac{1}{8};\ \ q = \frac{1}{2}\)
What's the intermediate step?
What I tried was:
(from the first equation) \(\displaystyle \large q = \sqrt{\frac{280}{a}-1}\)
and then substituting that for q in the second equation, but that gets very complex and I get stuck, it seems there is an easier way.
first term + third term = 280
fourth term + sixth term = 35
I need to find the ratio of the sequence and the first term, this problem is solved in the book in which it appears, but the solution says only:
q = ratio, a = first term
\(\displaystyle \large a + aq^2 = 280\\
aq^3 + aq^5 = 35\)
and thus
\(\displaystyle \large q^3 = \frac{1}{8};\ \ q = \frac{1}{2}\)
What's the intermediate step?
What I tried was:
(from the first equation) \(\displaystyle \large q = \sqrt{\frac{280}{a}-1}\)
and then substituting that for q in the second equation, but that gets very complex and I get stuck, it seems there is an easier way.