Tag: sum to infinite terms of a gp

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Questions Related to sum to infinite terms of a gp

Multiple choice sum to infinite terms of a gp sequence, progression and series maths

For first $n$ natural numbers we have the following results with usual notations $ \displaystyle \sum _{r=1}^{n}r =\frac{n(n+1)}{2}, \sum _{r=1}^{n}r^{2} =\frac{n(n+1)(2n+1)}{6},\sum _{r=1}^{n}r^{3}=\left ( \sum _{r=1}^{n}r \right )^{2}$ If $\displaystyle a _{1}a _{2}....a _{n} \in A.P $ then sum to $n$ terms of the sequence $\displaystyle \frac{1}{a _{1}a _{2}},\frac{1}{a _{2}a _{3}},...\frac{1}{a _{n-1}a _{n}}$ is equal to $\displaystyle \frac{n-1}{a _{1}a _{n}}$
 and the sum to $ n$ terms of a $G.P$ with first term '$a$' & common ratio '$r$' is given by  $\displaystyle S _{n}= \frac{lr-a}{r-1}$ for $ r \neq 1 $ for $ r =1 $ sum to $n$ terms of same $G.P.$ is $n$ $a$, where the sum to infinite terms of$G.P.$ is the limiting value of
 $\displaystyle \frac{lr-a}{r-1} $ when $\displaystyle n \rightarrow \infty ,\left |  r \right | < l $ where $l$ is the last term of $G.P.$  On the basis of above data answer the following questionsThe sum to infinite terms of the series $\displaystyle \frac{1}{2}+\frac{1}{6}+\frac{1}{18}+.. $ is equal to ?

  1. $\displaystyle \frac{4}{3}$
  2. $\displaystyle \frac{3}{4}$
  3. $\displaystyle \frac{8}{3}$

  4. Does not exit
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let, ${ S } _{ \infty  }=\dfrac { 1 }{ 2 } +\dfrac { 1 }{ 6 } +d\frac { 1 }{ 18 } +..\infty $

$\Rightarrow { S } _{ \infty  }=\dfrac { 1 }{ 2 } \left( 1+\dfrac { 1 }{ 3 } +\dfrac { 1 }{ { 3 }^{ 2 } } +....\infty  \right) $

As we know that, sum of infinite G.P series $=\dfrac { a }{ 1-r } $

Therefore, $ { S } _{ \infty  }=\dfrac { 1 }{ 2 } \left( \dfrac { 1 }{ 1-\left( 1/3 \right)  }  \right) =\dfrac { 3 }{ 4 } $

Ans: B

Multiple choice sum to infinite terms of a gp sequence, progression and series maths

If $\displaystyle x=\sum _{a=0}^{\infty }a^{n},y=\sum _{a=0}^{\infty }b^{n},z=\sum _{a=0}^{\infty }c^{n}$ Where $a,b,c $ are in A.P and $\displaystyle \left | a \right |<1,\left | b \right |<1,\left | c \right |<1$ then $x,y,z$ are in

  1. H.P

  2. Arithmetic-Geometric progression

  3. A.P

  4. G.P
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given $\displaystyle \left | a \right |< 1,\left | b \right |< 1,\left | c \right |< 1,\ \ a,b,c \in A.P$


and $\displaystyle \sum _{n=0}^{\infty }a^{n}=\frac{1}{1-a},\sum _{n=0}^{\infty }b^{n}=\frac{1}{1-b},\sum _{r=0}^{\infty }c^{n}=\frac{1}{1-c}$

$\displaystyle \therefore x=\frac{1}{1-a},y=\frac{1}{1-b},c=\frac{1}{1-c}$

$\displaystyle \Rightarrow a=\frac{x-1}{x},b=\frac{y-1}{y},c=\frac{z-1}{z}$

$\displaystyle \because 2b=a+c \ as \ a,b,c \in A.P$

$\displaystyle 2\left ( \frac{y-1}{y} \right )=\frac{x-1}{x}+\frac{z-1}{z}\Rightarrow \frac{2}{y}=\frac{1}{x}+\frac{1}{z}$

$\displaystyle \Rightarrow x,y,z \in H.P$

Multiple choice sum to infinite terms of a gp sequence, progression and series maths

If $R \subset\left ( 0,\pi  \right )$ denote the set of values of which satisfies the equation $ \displaystyle 2^{\left ( 1+\left | \cos x \right |+\left | cos^{2}x \right |+\left | cos^{3}x \right | \right )+\left | cos^{4}x  \right |...............\infty}=4$ then $R$ equals

  1. $\displaystyle\left \{ -\frac{\pi }{3} \right \}$
  2. $\displaystyle\left \{ \frac{\pi }{3},\frac{2\pi }{3} \right \}$
  3. $\displaystyle\left \{ \frac{-\pi }{3},\frac{2\pi }{3} \right \}$
  4. $\displaystyle\left \{ \frac{\pi }{3},\frac{-2\pi }{3} \right \}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

${ 2 }^{ \left( 1+\left| \cos { x }  \right| +\left| \cos ^{ 2 }{ x }  \right| +.........\infty  \right)  }={ 2 }^{ 2 }\ \Rightarrow 1+\left| \cos { x }  \right| +\left| \cos ^{ 2 }{ x }  \right| +.........\infty =2\ \Rightarrow \dfrac { 1 }{ 1-\left| \cos { x }  \right|  } =2\ \Rightarrow 1-\left| \cos { x }  \right| =\dfrac { 1 }{ 2 } \ \Rightarrow \left| \cos { x }  \right| =\dfrac { 1 }{ 2 } \ \Rightarrow x=\dfrac { \pi  }{ 3 } ,\dfrac { 2\pi  }{ 3 } $
  in the range $\left( 0,\pi  \right) $

Multiple choice sum to infinite terms of a gp sequence, progression and series maths

The sum of the series
$\dfrac { 1 } { 1.2 } - \dfrac { 1 } { 2.3 } + \dfrac { 1 } { 3.4 } \ldots \ldots \ldots$  up to  $\infty$  is equal to

  1. $\log _{ { { e } } } \left( \dfrac { 4 }{ { e } } \right) $
  2. $2 \log _ { e } 2$
  3. $\log _ { e } 2 - 1$
  4. $\log _ { e } 2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice sum to infinite terms of a gp sequence, progression and series maths

The sum of the infinite series, ${ 1 }^{ 2 }-\frac { { 2 }^{ 2 } }{ 5 } +\frac { { 3 }^{ 2 } }{ { 5 }^{ 2 } } -\frac { { 4 }^{ 2 } }{ { 5 }^{ 3 } } +\frac { { 5 }^{ 2 } }{ { 5 }^{ 4 } } -\frac { { 6 }^{ 2 } }{ { 5 }^{ 5 } } +.........$ is :

  1. $\frac { 1 }{ 2 } $
  2. $\frac { 25 }{ 24 } $
  3. $\frac { 25 }{ 54 } $
  4. $\frac { 125 }{ 252 } $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This is an arithmetico-geometric series of the form sum(n^2 * r^(n-1)). The sum can be found using the method of differences or differentiation of geometric series. The result for this specific series is 1/2.

Multiple choice sum to infinite terms of a gp sequence, progression and series maths

The first term of an infinitely decreasing G.P. is unity and its sum is S. The sum of the squares of the terms of the progression is

  1. $\displaystyle \frac {S}{2S-1}$
  2. $\displaystyle \frac {S^2}{2S-1}$
  3. $\displaystyle \frac {S}{2-S}$
  4. $S^2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let common ratio is $r<1$
Then G.P is $1,r,{ r }^{ 2 },{ r }^{ 3 },...\infty $
$S=1+r+{ r }^{ 2 }+{ r }^{ 3 }+...\infty $
$\displaystyle \Rightarrow S=\frac { 1 }{ 1-r } $
Then G.P formed by squaring the terms 
$1,{ r }^{ 2 },{ r }^{ 4 },{ r }^{ 6 },...\infty $
$\displaystyle { S }'=\frac { 1 }{ 1-{ r }^{ 2 } } =\frac { 1 }{ \left( 1-r \right) \left( 1+r \right)  } =\frac { { S }^{ 2 } }{ 2S-1. } $

Multiple choice sum to infinite terms of a gp sequence, progression and series maths

 If  $0<\phi < \pi /2,$   and
 $x= \sum _{n=0}^{\infty} \cos ^{2n} \phi$, $ y=\sum _{n=0}^{\infty } \sin ^{2n} \phi$                     
and $z=\sum _{n=0}^{\infty} \cos ^{2n} \phi \sin ^{2n} \phi $ 
then

  1. xyz $=$xz+y
  2. xyz$=$xy+z
  3. xyz$=$x+y+z
  4. xy$=$yz+z
Reveal answer Fill a bubble to check yourself
B,C Correct answer
Explanation

The series x converges to a value $\dfrac{1}{1-{\cos^2 \phi}}\;=\;{cosec^2 \phi}$

Similarly y converges to  a value ${sec^2 \phi} $

In a similar fashion.. z converges to $\dfrac{{cosec^2 \phi}{sec^2 \phi}}{{cosec^2 \phi}{sec^2 \phi} - 1}$
 $ z = \dfrac{xy}{xy - 1} $

$xy + z = xyz&gt;$; Option B.
And, also
$x + y = xy$
$\therefore xyz = x+ y+ z&gt;$; Option C.

Multiple choice sum to infinite terms of a gp sequence, progression and series maths

Find the sum of the infinite geometric series where the beginning term is $-1$ and the common ratio is $\dfrac{1}{2}$.

  1. $1$
  2. $-1$
  3. $2$
  4. $-2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given that first term $a=-1$ and common ratio is $r=\dfrac{1}{2}$

We know $\text{sum} = \dfrac{a}{1-r}$
$\Rightarrow \text{sum} = \dfrac{-1}{1-\frac{1}{2}}$
$\Rightarrow \text{sum} = \dfrac{-1}{\frac{1}{2}}$
$\Rightarrow \text{sum} = -2$

Multiple choice sum to infinite terms of a gp sequence, progression and series maths

$1 + x + x^2 + x^3 +......$ = ?

  1. $\dfrac{1}{1-x}$
  2. $\dfrac{1}{1-x^2}$
  3. $\dfrac{1}{1-x^3}$
  4. $\dfrac{x}{1-x}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Sum of a GP $a,ar,ar^2......$

$=\dfrac{a(r^n -1)}{r-1}$    (for $n$ terms)
For the given series,
$a=1, r=x, n \to \infty$
Sum of the series is:
$\text{sum} = \lim _{n \to \infty} \dfrac{x^n-1}{x-1}$
For $x>1$ 
$\text{sum} \to \infty$
For $x<1$
$x^n \to 0$
$\Rightarrow \text{sum} = \dfrac{-1}{x-1}$
$\Rightarrow \text{sum} = \dfrac{1}{1-x}$

Multiple choice sum to infinite terms of a gp sequence, progression and series maths

If $a=\sum _{ n=0 } ^{\infty  }{x^n } ,b=\sum _{n=0  }^{ \infty  }{ y^n } , c=\sum _{n=0  }^{ \infty  }{ (xy)^n } $ where $|x| ,| y| < 1$ ; then

  1. $abc = a + b + c$
  2. $ab + bc = ac + b$
  3. $ac + bc = ab + c$
  4. $ab + ac = bc + a$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Clearly every summation is infinite series,
$a=\cfrac{1}{1-x}, b = \cfrac{1}{1-y}$ and $c=\cfrac{1}{1-xy}$
or $x=1-\cfrac{1}{a}, y=1-\cfrac{1}{b}$
Simplifying above equation we get, $ac+bc=ab+c$