Tag: properties of an ap

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

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

  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
Reveal answer Fill a bubble to check yourself
D Correct answer
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 

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

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

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

  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
Reveal answer Fill a bubble to check yourself
C Correct answer
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.
Multiple choice maths arithmetic progressions complete the a.p series with given information properties of an ap problems on 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.

  1. equal

  2. unequal

  3. different

  4. various
Reveal answer Fill a bubble to check yourself
A Correct answer
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 

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

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?

  1. A.P.

  2. G.P.

  3. H.P.

  4. AGP
Reveal answer Fill a bubble to check yourself
A Correct answer
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 

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

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

  1. $8$
  2. $11$
  3. $12$
  4. $13$

  5. None of these
Reveal answer Fill a bubble to check yourself
D Correct answer
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 $=13$

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

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

  1. $2$
  2. $3$
  3. $4$
  4. $5$
Reveal answer Fill a bubble to check yourself
B Correct answer
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

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

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

  1. $!8$th
  2. $15$th
  3. $28$th
  4. $27$th
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $3$).

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

${ T } _{ m }$ 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 $m=$

  1. $3$
  2. $6$
  3. $9$
  4. $12$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

T_m = mC3. The condition T_{m+1} - T_m = 15 becomes (m+1)C3 - mC3 = 15, which simplifies to mC2 = 15. Solving m(m-1)/2 = 15 gives m^2 - m - 30 = 0, so (m-6)(m+5)=0. Thus m=6.