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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.

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

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

  1. Arithmetic progression

  2. Geometric progression

  3. Harmonic progression

  4. Arithmetico-Geometric progression
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

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

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.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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.