(a) If 3, x, y, 18 are the terms of an Arithmetic Progression (A.P), find the values of x and y. (b)(i) The sum of the second and third terms of a grometric...

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(a) \(3,\ x,\ y,\ 18\) are consecutive terms of an A.P. The first term is \(3\) and the fourth term is \(18\):

\[18=3+(4-1)d=3+3d\Rightarrow d=5\]

So \(x=3+5=8\) and \(y=8+5=13\).

\(x=8,\ y=13\)

(b)(i) For a G.P. with first term \(a\) and ratio \(r\), terms are \(a,\ ar,\ ar^{2},\ ar^{3},\ldots\)

"Sum of second and third terms \(=6\times\) fourth term":

\[ar+ar^{2}=6ar^{3}\]

Divide through by \(ar\ (a,r\neq0)\): \(1+r=6r^{2}\Rightarrow 6r^{2}-r-1=0\).

\[r=\frac{1\pm\sqrt{1+24}}{12}=\frac{1\pm5}{12}\]

\(r=\tfrac{1}{2}\) or \(r=-\tfrac{1}{3}\).

(ii) The ratio is positive, so \(r=\tfrac{1}{2}\). The second term is \(ar=8\):

\[a\times\tfrac{1}{2}=8\Rightarrow a=16\]

First six terms: \(16,\ 8,\ 4,\ 2,\ 1,\ \tfrac{1}{2}\).