How many terms are there in the following sequences? 1, 2, 4, 8, ..,...............,., 4096
13
Given:
The sequence 1, 2, 4, 8, ..................., 4096 is a geometric progression.
Concept:
A geometric sequence: A sequence in which each term is obtained by multiplying or dividing a fixed number by the preceding number is known as a geometric sequence.
Sequence = a, ar, ar2,.........ar(n-1)
Here, a = first term, r = common ratio, n = position of the term
Let the sequence a1, a2, a3,........an be a geometric progression.
- Common ratio (r) = successive term/previous term = a2/a1 = .... = an/a(n-1)
- General term (nth term) = a n = ar(n-1)
Calculation:
Since we know that the general term (nth term) = a n = ar(n-1)
Here, a = 1, r = 2 and an = 4096
⇒ a n = ar(n-1)
⇒ 4096 = 1 × 2(n-1)
⇒ 212 = 2(n-1) [If am = an then, m = n]
⇒ 12 = n-1
⇒ n = 13
∴ There are 13 terms in the given sequence.
Additional Information
- The sum of n terms of a geometric series = Sn = a(1 – rn)/(1 – r) if r < 1
- The sum of n terms of a geometric series = S n = a(rn -1)/(r – 1) if r > 1
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
