(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...
Question 1 Report
(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 progression is six times the fourth term. Find the two possible values of the common ratio.
(ii) If the second term is 8 and the common ratio is positive, find the first six terms.
Answer Details
(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: \(16,\ 8,\ 4,\ 2,\ 1,\ \tfrac{1}{2}\).
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