### Properties of geometric sequence

Question Sample Titled 'Properties of geometric sequence'

Let ${a}$ , ${b}$ and ${c}$ be positive constants. If ${a}{x}^{{2}}+{2}{b}{x}+{c}={0}$ has repeated roots, which of the following must be true?

 I. ${{\log{{a}}}^{{4}}}$ , ${{\log{{b}}}^{{4}}}$ and ${{\log{{c}}}^{{4}}}$ form an arithmetic sequence. II. ${3}{a}$ , ${3}{b}$ and ${3}{c}$ form a geometric sequence. III. ${a}+{4}$ , ${b}+{4}$ and ${c}+{4}$ form a geometric sequence.

A
I and II only
B
I and III only
C
None of the above
D
II only

 ∵ ${a}{x}^{{2}}+{2}{b}{x}+{c}$ $={0}$ has repeated roots. ∴ $\Delta$ $={0}$ ${\left({2}{b}\right)}^{{2}}-{4}{a}{c}$ $={0}$ ${b}^{{2}}$ $={a}{c}$ ∴  ${a}$ , ${b}$ and ${c}$ form a geometric sequence. ∴  ${3}{a}$ , ${3}{b}$ and ${3}{c}$ form a geometric sequence.  ∵  ${a}$ , ${b}$ and ${c}$ form a geometric sequence. ∴  ${\log{{a}}}$ , ${\log{{b}}}$ and ${\log{{c}}}$ form an arithmetic sequence. ∴  ${4}{\log{{a}}}$ , ${4}{\log{{b}}}$ and ${4}{\log{{c}}}$ form an arithmetic sequence. ∴  ${{\log{{a}}}^{{4}}}$ , ${{\log{{b}}}^{{4}}}$ and ${{\log{{c}}}^{{4}}}$ form an arithmetic sequence.  ${a}$ , ${b}$ and ${c}$ form a geometric sequence. This does not provide that ${a}+{4}$ , ${b}+{4}$ and ${c}+{4}$ form a geometric sequence.

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