Tag: maths

Questions Related to maths

In an arithmetic progression the sum of two terms equidistant from the beginning and the end is always _____ to the sum of the first and last terms.

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

  1. equal

  2. unequal

  3. different

  4. various

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Correct Option: A
Explanation:

Let $a$ be first term $d$ be common difference of an AP having $n$ number of terms.

So $m$th term from the beginning is, $a _m=a+(m-1)d$
and $m$ the term from end is $a _{n-m}= a+(n-1)d-(m-1)d=a+(n-m)d$
So $a _m+a _{n-m}=2a+(n-1)d=[a]+[a+(n-1)d]=$ sum of first and last term 

$\dfrac{1}{x},\dfrac{2}{x},\dfrac{3}{x},....$ is a property of

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

  1. G.P.

  2. A.P.

  3. A.S.

  4. H.P.

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Correct Option: B
Explanation:

$\dfrac{1}{x},\dfrac{2}{x},\dfrac{3}{x},....$ is a property of A.P.
Here the constant x is divided from each term of an A.P. with same common difference.

In which property the sum of two terms equidistant from the beginning and the end is always same or equal to the sum of the first and last terms?

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. AGP

Show answer
Correct Option: A
Explanation:

Let $a$ be first term $d$ be common difference of an AP having $n$ number of terms.

So $m$th term from the beginning is, $a _m=a+(m-1)d$
and $m$ the term from end is $a _{n-m}= a+(n-1)d-(m-1)d=a+(n-m)d$
So $a _m+a _{n-m}=2a+(n-1)d=[a]+[a+(n-1)d]=$ sum of first and last term 

Hence option 'A' is correct choice 

How many natural numbers are there between $23$ and $100$ which are exactly divisible by $24$?

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

  1. $8$

  2. $11$

  3. $12$

  4. $13$

  5. None of these

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Correct Option: D
Explanation:

Required numbers are $24,30,36,40,.....96$
This is an A.P. in which $a=24,d=6,l=96$
Let the number of terms in it be $n$.
Then ${t} _{n}=96$ $\Rightarrow$ $a+(n-1)d=96$
$\Rightarrow$ $24+(n-1)\times 6=96$
$\Rightarrow$ $(n-1)=12$
$\Rightarrow$ $n=13$
Required number of numbers $=13$

The sum of first $10$ terms and $20$ terms of an AP are $120$ and $440$ respectively. What is the first term?

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

  1. $2$

  2. $3$

  3. $4$

  4. $5$

Show answer
Correct Option: B
Explanation:

Let the first term be $a$ and common difference be $d$.
So, sum of first $10$ terms $=\dfrac { 10 }{ 2 } (2a+(10-1)d)$

$\implies 120 =5(2a+9d)$
$\implies 24=2a+9d$ .............. $(i)$
Sum of first 20 terms $=\frac { 20 }{ 2 } (2a+(20-1)d)$
$\implies 440 =10(2a+19d)$
$\implies 44=2a+19d$ ......... $(ii)$
Subtracting equation (i) from (ii) gives
$20=10d$
$\implies d=2$
Common difference =2
Substituting in $(i)$, we get 
$a=3$
Hence, option B is correct

$T _m$ denotes the number of Triangles that can be formed with the vertices of a regular polygon of $m$ sides.If $T _m+ _1-T _m=15$ , then $m$

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

  1. 3

  2. 6

  3. 9

  4. 12

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Correct Option: B
Explanation:

$T _m+ _1-T _m=15$ 

$\Rightarrow ^{(m+1)}C _3-^{(m)}C _3=15$

By verification $m=6$ by solve

Which term of A.P. $20, 19\displaystyle\frac{1}{4}, 18\frac{1}{2}$,..... is first negative term?

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

  1. $!8$th

  2. $15$th

  3. $28$th

  4. $27$th

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Correct Option: C
Explanation:

$20, \displaystyle 19\frac{1}{4}, 18\frac{1}{2}, ....$
or $\displaystyle 20, \frac{77}{4}, \frac{37}{2}, ....$
$a=20$
$d=\displaystyle\frac{77}{4}-20=\frac{-3}{4}$
Let $n^{th}$ term of A.P. be first negative term
So, $20+(n-1)\left(\displaystyle\frac{-3}{4}\right)<0$
$\Rightarrow 80-3n+3<0$
$\Rightarrow 3n>83$
$\Rightarrow n > 27\displaystyle\frac{2}{3}$
Hence, $28^{th}$ term is first negative term.
(Option $3$).

${ T } _{ m }$ denotes the number of triangles that can be formed with the vertices of a regular polygon of m sides. If ${ { T } _{ m+1 } }-{ { T } _{ m } }=15,$ then $m=$

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

  1. $3$

  2. $6$

  3. $9$

  4. $12$

Show answer
Correct Option: B

If $a,b,c$ are distinct and the roots of $\left( b-c \right) { x }^{ 2 }+\left( c-a \right) x+\left( a-b \right) =0$ are equal, then $a,b,c $ are in

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

  1. Arithmetic progression

  2. Geometric progression

  3. Harmonic progression

  4. Arithmetico-Geometric progression

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Correct Option: A
Explanation:

Clearly $x=1$ is a solution
$\therefore$  product of the roots $=\dfrac { a-b }{ b-c }$ 
$\therefore \left( 1 \right) \left( 1 \right) =\dfrac { a-b }{ b-c }$ 
$\Longrightarrow b-c=a-b$
$\Longrightarrow2b=a+c\Longrightarrow a,b,c$ are in Arithmetic progression.

Say true or false.
In an $A.P$., sum of terms equidistant from the beginning and end is constant and is equal to the sum of the first and last term.

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

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

Consider mth term of an AP 
$t _m=a+(m-1)d$
Now, consider (n-m)th term:
$t _{n-m}=a+(n-m-1)d$
Sum $= 2a+(n-1)d $

$= a +a+(n-1)d $
$= a+l$
Therefore, it is true that Sum of the terms is equal to the sum of the first and last terms.