Four numbers are in A.P. The first, the second and the fourth are in G.P. Find the numbers if the common difference is equal to the common ratio.
Let the terms of the A.P. be $a_1,a_1+d,a_1+2d,a_1+3d$ and the terms of the G.P. will be $a_1,a_1+d,a_1+3d.$ We know that $d=q.$ I am not sure what to do from here.
Best Answer
Write $\dfrac {a_1+d}{a_1} = \dfrac{a_1+3d}{a_1+d} = d$.
Now we also have $\dfrac {d}{a_1} = \dfrac {2d}{2a_1} =\dfrac{2d}{a_1+d} = d-1$.
This gives $a_1 = d$, and from $\dfrac {a_1+d}{a_1}=d$ we have $a_1=d=2$.