I came across a question today
Given a sequence of four numbers such that the first three are in G.P. and the last three are in A.P. with common difference $6$. If the first and the fourth number are equal, then common ratio of the G.P. is?
Now correct way of doing this is given as:
Let the last three numbers be $a, a+6, a+12$, so that the first three numbers are $a+12, a, a+6$. As these are in G.P. $$a^2 = (a+12)(a+6) \Rightarrow a=-4$$
Common ratio is $-2$
But how I did this is
I took first term of G.P. as $a_1$ and the first term of A.P. as $a_2$. So the series is $$a_1, (a_1r) \text{ or }(a_2), (a_1r^2) \text{ or } (a_2+6), a_2 +12$$
So, $a_1 = a_2+12$ and $r= \dfrac{a_2+6}{a_2}$ and $a_1r^2 – a_1r=6$
Solving these three equations I get Common ratio (r) = $-0.5$
What is wrong in my solution? I know that my solution is bit more time consuming but why is it wrong?
Best Answer
Let the four terms be: $$\begin{align} &a\\ &ar=&&a-12\\ &ar^2=&&a-6\\ & &&a\end{align}$$ From the above, we have $$6=a(1-r^2)\\ 12=a(1-r)$$ Dividing (allowed as $r\neq 1$) and solving gives $$r=-\frac12\quad\blacksquare$$
Solving gives $a=8$, hence the four numbers are $$\lbrace 8,-4,\;2,\;8\rbrace$$