Tag: complete the a.p series with given information

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

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

  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. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation cos(px) + cos(qx) = 0 implies 2cos((p+q)x/2)cos((p-q)x/2) = 0. Solving for x gives the roots in AP with the specified common differences.

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

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

  1. $\dfrac{n-1}{2}(3n+2)$
  2. $3+\dfrac{n-1}{2}(3n+2)$
  3. $1+\dfrac{n-1}{2}(3n+2)$
  4. $3+\dfrac{(n+1)(3n+2)}{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The sequence is 3, 7, 14, 24. The differences are 4, 7, 10. This is a quadratic sequence. Testing n=1 gives 3, n=2 gives 7. Option B fits the pattern.

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

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

  1. A.P

  2. G.P

  3. H.P

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

By calculating the distances from A to the lines, one can show that the segments form an arithmetic progression based on the properties of the lines and the point A.

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

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

  1. $49^2-1$
  2. $49^2$
  3. $50^2+1$
  4. $49^2+2$
Reveal answer Fill a bubble to check yourself
D Correct answer
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.

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

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

  1. 1

  2. a+b+c

  3. m+n+p

  4. 0
Reveal answer Fill a bubble to check yourself
D Correct answer
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,
$m(b-c)+n(c-a)+p(a-b)=m(n-p)d+n(p-m)d+p(m-n)d$
$=d(mn-mp+np-nm+pm-pn)$
$=d \times 0=0$
Multiple choice maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

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 :

  1. 1 and 2 only

  2. 2 and 3 only

  3. 1 and 3 only

  4. 1, 2 and 3 only
Reveal answer Fill a bubble to check yourself
D Correct answer
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.

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

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

  1. $32$
  2. $33$
  3. $34$
  4. $35$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $32$.

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

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

  1. $\displaystyle \frac { 1 }{ mn } $
  2. $\displaystyle \frac { m }{ n } $
  3. $\displaystyle 1$
  4. $\displaystyle \frac { n }{ m } $
Reveal answer Fill a bubble to check yourself
C Correct answer
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 } $

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