Given a geometric sequence with t5= 162 and t8= -4374, determine the first 3 terms of the sequence. ?

Question

Given a geometric sequence with t5= 162 and t8= -4374, determine the first 3 terms of the sequence. ?

Answer 1

In a geometric sequence, the ratio of successive terms is the same.  You need to know this ratio before you can do much else.  Call it r.  In that case, t6 = (t5)r, t7 = t6(r) = t5(r^2), and t8 = t7(r) = t6(r^2) = t5(r^3).

So, -4374 = 162(r^3)

r^3 = -4374/162 = -27

So r = -3

Each term in the sequence is (-3) times the last one.

Now - perhaps it's obvious at this point that t5 = t1(r^4).  If not, start with t5 and trace it backwards as I went forwards to t8.  That means

t1 = t5/(r^4) = 162/(-3)^4 = 2

So your first three terms are 2, -6, 18.  QED