Given a geometric sequence where the $5$th term $= 162$ and the $8$th term $= -4374$, determine the first three terms of the sequence.
I am unclear how to do this without being given the first term or the common ratio. please help!!
[Math] Finding the first three terms of a geometric sequence, without the first term or common ratio.
sequences-and-series
Best Answer
We have that $162 = a_{1}r^{4}$ and $-4374 = a_{1}r^{7}$ by the formula $a_{n} = a_{1}r^{n-1}$.
Then solving for $a_{1}$ in both equations and setting them equal to one another, $$\frac{162}{r^{4}} = \frac{-4374}{r^{7}}$$
You can then solve for $r$ (your common ratio), and subsequently $a_{1}$ (your first term). You then have all of the information you need.