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!!
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$\begingroup$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.
$\endgroup$ 1 $\begingroup$If $ar^4 = 162$ and $ar^7 = 4374$ then $$ \frac{-4374}{162} = \frac{ar^7}{ar^4} = r^3 $$ so $$ r^3 = \frac{-4374}{162} = -27. $$ If you know $r^3=-27$ can you find $r$? If you know $r$ and $ar^4$ can you find $a$?
$\endgroup$ $\begingroup$I had a similar question and tried to solve it alone i used this way
since the initial value is f(1) so u use this equation f(n) = ar^n-1
since u have f(5) as initial value or actually (known value) then we have to change it to this
f(n) = a5 * r (n-5) , so we know that the 8 th term f(8) = -4734 so : f(8) = a5 * r(8-5) and f(5)= 162
-4734 = 162 *r^3
-47324/162 = r^3
r^3 = -27 then calculate ......
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