Tag: complete the a.p series with given information

Questions Related to complete the a.p series with given information

$a$ should be $5$ to make the series $2,a,8,...$ to be in AP

maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

  1. True

  2. False

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Correct Option: A
Explanation:

Given series $2,a,8$

The condition to be in AP is 
$2b=a+c\2a=2+8\2a=10\a=5$

If the solution as $\cos p\theta +\cos q\theta=0$ are in $AP$ then the common difference is

maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

  1. $\dfrac {\pi}{p+q}$ or $\dfrac {\pi}{p-q}$

  2. $\dfrac {\pi}{p+q}$ or $\dfrac {\pi}{2(p-q)}$

  3. $\dfrac {\pi}{2(p+q)}$ or $\dfrac {\pi}{2(p-q)}$

  4. $None\ of\ these$

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Correct Option: A

The $n^{th}$ term of the series $3+7+14+24+.....$ is

maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

  1. $\dfrac{n-1}{2}(3n+2)$

  2. $3+\dfrac{n-1}{2}(3n+2)$

  3. $1+\dfrac{n-1}{2}(3n+2)$

  4. $3+\dfrac{(n+1)(3n+2)}{2}$

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Correct Option: B

A line passes through the variable point $A(\lambda +1,2\lambda)$ meets the lines $7x+y-16=0,\ 5x-y-8=0,\ x-5y+8=0$ at $b,c,d$ respectively. Then $AC, AB, AD$ are in

maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

  1. A.P

  2. G.P

  3. H.P

  4. None of these

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Correct Option: A

If $T _n$ denotes the nth terms of the series.$ 2+3+6+11+18+..........$ , then $T _50$ is :

maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

  1. $49^2-1$

  2. $49^2$

  3. $50^2+1$

  4. $49^2+2$

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Correct Option: D
Explanation:

$2+3+6+11+18+.................$

$\begin{matrix} { T _{ n } }=a{ x^{ 2 } }+bx+c \ { T _{ 1 } }=a+b+c=2 \ { T _{ 2 } }=a+b+c=3 \ { T _{ 3 } }=a+b+c=6 \ 3a+b=1 \ 5a+b=3 \ 2a=2 \ a=1 \ b=-2 \ c=3 \  \end{matrix}$
${T _n} = {n^2} - 2n + 3$
${T _{50}} = {\left( {50} \right)^2} - 2\left( {50} \right) + 3$
${T _{50}} = {\left( {49} \right)^2} + 2$
Hence,
Option $D$ is correct answer.

If the m+n, n+p, p+n terms of an AP are a, b, c respectively, then m(b-c)+n(c-a)+p(a-b) is

maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

  1. 1

  2. a+b+c

  3. m+n+p

  4. 0

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Correct Option: D
Explanation:
$T _{m}=A+(m-1)d=a$
$T _{n}=A+(n-1)d=b$
$T _{p}=A+(p-1)d=c$
From these equations, we get,
$a-b=(m-n)d$
$b-c=(n-p)d$
and $c-a=(p-m)d$
Now,
$m(b-c)+n(c-a)+p(a-b)=m(n-p)d+n(p-m)d+p(m-n)d$
$=d(mn-mp+np-nm+pm-pn)$
$=d \times 0=0$

Let $a,b,c$ be in AP and $k\neq 0$ be a real number. WHich of trhe following are correct ?
1.$ka,kb,kc$ are in Ap
2. $k-a,k-b,k-c$ are in AP
3. $\dfrac{a}{k},\dfrac{b}{k},\dfrac{c}{k}$ are in AP
Select the correct answer using the code given below :

maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

  1. 1 and 2 only

  2. 2 and 3 only

  3. 1 and 3 only

  4. 1, 2 and 3 only

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Correct Option: D
Explanation:

$a, b, c \in A.P. \quad \left( \text{Given} \right)$

Therefore,
$(1) \quad\text{when a constant term is multiplied to each term of an A.P. then the resultant is also an A.P.}$
$ka, kb, kc \in A.P., \quad k \ne 0$

$(2)\quad\text{when a constant is subtracted from each term of an A.P. then the resultant is also an A.P.}$
$\Rightarrow k - a, k-b, k - c \in A.P.$

$(3)\quad\text{when a constant is divided from each term of an A.P. then the resultant is also an A.P.}$
$\Rightarrow \cfrac{a}{k}, \cfrac{b}{k}, \cfrac{c}{k} \in A.P.$
Hence all statements are correct.

Find the next term of the series:
$22, 26, 29, 31,$ ..........

maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

  1. $32$

  2. $33$

  3. $34$

  4. $35$

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Correct Option: A
Explanation:

Lets find the difference between the numbers of the series:

$22$       $26$       $29$       $31$       $x$.....
      $4$          $3$         $2$        $1$
Since, the difference is in A.P.
$\therefore x=31+1=32$
Hence, the answer is $32$.

$\displaystyle a^{2}\left ( b+c \right ),b^{2}\left ( c+a \right ),c^{2}\left ( a+b \right )$ provided $\displaystyle \sum ab\neq 0.$ if $a=b=c$ then above series is in:

maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

  1. AP

  2. GP

  3. HP

  4. AGP

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Correct Option: A,B,C,D

If the $m^{th}$ term and the $n^th$ term of an AP are respectively $\displaystyle \frac { 1 }{ n } $ and $\displaystyle \frac { 1 }{ m } $, then the $mn^{th}$ term of the AP is

maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

  1. $\displaystyle \frac { 1 }{ mn } $

  2. $\displaystyle \frac { m }{ n } $

  3. $\displaystyle 1$

  4. $\displaystyle \frac { n }{ m } $

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Correct Option: C
Explanation:

Let a and d be the first term and common difference of an AP.
Since, $\displaystyle { T } _{ m }=\frac { 1 }{ n } $
$\displaystyle \therefore a+\left( m-1 \right) d=\frac { 1 }{ n } $....(i)
and $\displaystyle { T } _{ n }=\frac { 1 }{ m } $
$\displaystyle \Rightarrow a+\left( n-1 \right) d=\frac { 1 }{ m } $.....(ii)
On solving Eqs. (i) and (ii), we get
$\displaystyle a=\frac { 1 }{ mn } and\quad d=\frac { 1 }{ mn } $
$\displaystyle \therefore \quad { T } _{ mn }=a+\left( mn-1 \right) d$


$\displaystyle =\frac { 1 }{ mn } +\frac { \left( mn-1 \right)  }{ mn } $

$\displaystyle =\frac { mn }{ mn } =1$