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.
Similar Questions
What is the greatest value of the positive integer n satisfying the condition \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots + \frac{1}{{{2^{{\rm{n}} - 1}}}} < 2 - \frac{1}{{1000}}?\)
Let t1, t2, t3 ... be in GP. What is \(\rm \left(t_1 t_3 \ldots t_{21}\right)^{\frac{1}{11}}\) equal to ?
The geometric mean of a set of observations is computed as 10. The geometric mean obtained when each observations x iis replaced by \(\rm 3x_i^4\) is
If β, 2, 2m are in GP, then what is the value of \({\rm{\beta }}\sqrt {\rm{m}} \) ?
How many geometric progression is/are possible containing 27, 8 and 12 as three of its/their terms?
What is the value of α?
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
If p, q, r are in one geometric progression and a, b, c are in another geometric progression, then ap, bq, cr are in
Important Questions from Geometric Progressions
What is the greatest value of the positive integer n satisfying the condition \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots + \frac{1}{{{2^{{\rm{n}} - 1}}}} < 2 - \frac{1}{{1000}}?\)
The minimum value of the sum of real numbers a-5, a-4, 3a-3, 1, a8 and a10 with a > 0 is:
The arithmetic mean, geometric mean and median of six positive numbers a, a, b, b, c, c where a < b < c are \(\frac 7 3,\) 2, 2 respectively. Then what is the sum of the squares of all the six numbers?
If the side of a right angle triangle are a, ar, ar2 (r < 1), then r2 is equal to
In a geometric progression, first term is 7, the last term is 448 and the sum is 889. The common ratio of the geometric progression is
