Tag: complete the a.p series with given information

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

Which one is an example of A.P. property?

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

  1. Constant $a$ is added to each term of an A.P. will form a new A.P with different common difference

  2. Constant $a$ is subtracted to each term of an A.P. will form a new A.P with different common difference

  3. Constant $a$ is divided to each term of an A.P. will not form a new A.P with same common difference

  4. Constant $a$ is added to each term of an A.P. will form a new A.P with same common difference

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Correct Option: D
Explanation:

If you will subtract or add a constant number to each terms of an AP, then later sequence will also be in AP with same common difference

Hence option 'D' is correct choice 

Identify the property of A.P. used in the sequence: 

$(3 - x), (5 - x), (7 - x), (9 - x)$

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

  1. $3$ is a constant subtracted from the sequence

  2. $-x$ is a constant subtracted from the sequence

  3. $x$ is a constant subtracted from the sequence

  4. Number is a constant subtracted from the sequence

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Correct Option: C
Explanation:
Given sequence is $(3-x),(5-x),(7-x),(9-x)$
Here $3,5,7,9$ are in AP with common difference $2$
Using the property of AP, if we subtract the same no. from each term, then it will remain in AP.
i.e. $3-x , 5-x, 7-x, 9-x$ is all equal to $2$ which will be in AP.

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

Show answer
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

Show answer
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