Consider the following statements: 1. If each term of a GP is multiplied by same non-zero number, then the resulting sequence is also a GP. 2. If each term of a GP is divided by same non-zero number, then the resulting sequence is also a GP. Which of the above statements is/are correct?
Both 1 and 2
Concept:
Multiplying or dividing by a non-zero number does not change the common ratio of a G.P.
Calculation:
Multiplying or dividing by a non-zero number does not change the common ratio of a G.P.
Let the sequence a1, a2, a3 …. an be a G.P.
Common ratio = r = \(\rm a_2\over a_1\) = \(\rm a_3\over a_2\) = \(\rm a_n\over a_{n-1}\)
If each term is multiplied/divided by the same non-zero number, the r value will remain the same. Thus, the sequence will not change.
Important Questions from Geometric Progressions
If \(2^{\frac{1}{c}}, 2^{\frac{b}{a c}}, 2^{\frac{1}{a}}\) are in GP, then which one of the following is correct ?
If G is the geometric mean of numbers 1, 2, 22, 23,.....2n-1, then what is the value of 1 + 2log2G ?
If m is the geometric mean of \({\left( {\frac{{\rm{y}}}{{\rm{z}}}} \right)^{\log \left( {{\rm{yz}}} \right)}},{\rm{\;}}{\left( {\frac{{\rm{z}}}{{\rm{x}}}} \right)^{\log \left( {{\rm{zx}}} \right)}}{\rm{\;and\;}}{\left( {\frac{{\rm{x}}}{{\rm{y}}}} \right)^{\log \left( {{\rm{xy}}} \right)}}\) then what is the value of m?
The value of the infinite product \({6^{\frac{1}{2}}} \times {6^{\frac{1}{2}}} \times {6^{\frac{3}{8}}} \times {6^{\frac{1}{4}}} \times \ldots \) is
The geometric mean of the observations x 1, x 2, x 3, … x nis G 1. The geometric mean of the observations y 1, y 2, y 3,… y nis G 2. The geometric mean of observations \(\frac{{{{\rm{x}}_1}}}{{{{\rm{y}}_1}}},\frac{{{{\rm{x}}_2}}}{{{{\rm{y}}_2}}},\frac{{{{\rm{x}}_3}}}{{{{\rm{y}}_3}}}, \ldots \frac{{{{\rm{x}}_{\rm{n}}}}}{{{{\rm{y}}_{\rm{n}}}}}\) is
