[Math] Geometric sequence, finding the first term using only the sum, the number of terms and value of one term.

Question

In Geometric series: S = 56, a(2) = 16 and n = 3

S – sum, a(2) – second term, n – number of terms

Is it possible to get a(2) and a(3) from here? (If yes, hints would be awesome)

Thank You!

Answer 1

So we have $S = a_1 + a_2 + a_3$, as we are considering a geometric sequence, we have $a_2 = ra_1$, $a_3 = r^2a_1$. The first one gives $a_1 = r^{-1}a_2$, plugin this into the second gives us $a_3 = ra_2$. So $$ S = r^{-1}a_2 + a_2 + ra_2 = \left(\frac 1r + 1 + r\right)a_2 $$ From this we can compute $$ \frac 1r + 1 + r = \frac S{a_2} $$ hence $$ 1 + r + r^2 = \frac {Sr}{a_2} $$ which is a quadratic equation for $r$. I'm sure you can do it from here.