Tag: numbers and sequences

Questions Related to numbers and sequences

A series is:

maths numbers and sequences series introduction to series introduction to sequences and series

  1. A number of events, objects, or people of a similar or related kind coming one after another.

  2. Combination of terms following a particular pattern.

  3. Both A and B

  4. Non of the above

Show answer
Correct Option: C
Explanation:

A series is a number of events, objects, or people of a similar or related kind coming one after another and combination of terms following a particular pattern.

Series can be defined as:

maths numbers and sequences series introduction to series introduction to sequences and series

  1. a number of things or events that are arranged or happen one after the other.

  2. a set of regularly presented television shows involving the same group of characters or the same subject.

  3. set of books, articles, etc., that involve the same group of characters or the same subject.

  4. All of the above

Show answer
Correct Option: D
Explanation:
Series is defined as a number of things or events that are arranged or happen one after the other. A set of regularly presented television shows involving the same group of characters or the same subject (as the number of episode in increasing by $1$ each day). Set of books, articles, etc., that involve the same group of characters or the same subject.( as the books, article have common subject).

Ans: All of the above

A divergent series:

maths numbers and sequences series introduction to series introduction to sequences and series

  1. The infinite sequence of the partial sums of the series does not have a finite limit.

  2. $2+4+6+8+......$

  3. Both A and B are correct

  4. Only A is correct

Show answer
Correct Option: C
Explanation:
A divergent series with infinite sequence of the partial sums of the series does not have a finite limit.
Also, $2+4+6+8+...$ is approaching to infinity.
Hence both $A$ and $B$ statements are correct.

Which of the following option will complete the given series $1,6,15,?,45,66,91$?

maths numbers and sequences series introduction to series introduction to sequences and series

  1. $25$

  2. $26$

  3. $27$

  4. $28$

Show answer
Correct Option: D
Explanation:
Sol :- $ 1,6,15,?,45,66,91 $
$ 1\times 1 = 1 $
$ 3\times 2 = 6 $
$ 5\times 3 = 15 $
$ 7\times 4 = 28 $
$ 9\times 5 = 45 $
$ 11\times 6 = 66 $
$ 13\times 7 = 91 $ 
Ans $ d = 28 $ 

Select the most appropriate option to identify the INCORRECT number in the series.  $3,5,13,43,176,891,5353$

maths numbers and sequences series introduction to series introduction to sequences and series

  1. $5$

  2. $13$

  3. $43$

  4. $176$

Show answer
Correct Option: D

If $\left| x \right| <1$ and $\left| y \right| <1$, the sum to infinity of the series $x+y,({ x }^{ 2 }+xy+{ y }^{ 2 }),({ x }^{ 3 }+{ x }^{ 2 }y+x{ y }^{ 2 }+{ y }^{ 3 }),.........$ is

maths numbers and sequences series introduction to series introduction to sequences and series

  1. $\frac { x+y-xy }{ 1-x-y+xy } $

  2. $\frac { x+y+xy }{ 1-x-y+xy } $

  3. $\frac { x }{ 1-x } +\frac { y }{ 1-y } $

  4. $\frac { (x-y)(x+y-xy) }{ 1-x-y+xy } $

Show answer
Correct Option: A

Sum the following series to n terms: $3+5+9+15+23+...$

maths numbers and sequences series introduction to series introduction to sequences and series

  1. $\dfrac{n}{3}(n^{2}-8)$

  2. $\dfrac{n}{3}(n^{2}+8)$

  3. $\dfrac{n}{2}(n^{3}+8)$

  4. None of these

Show answer
Correct Option: B
Explanation:
We have 

$3+5+9+15+23+............+T _{n-1}+T _{n}$

The difference between the successive terms are $S$ difference are in $A.P$

Let $S _{n}$ denote the sum to $n$ terms of the given series 

Then, $S _{n}=3+5+9+15+23+........+ T _{n-1}+T _{n}..... (1)$

Also $S _{n}=3+5+9+15+.......+T _{n-1}+T _{n}........ (2)$

Subtracting $(2)$ from $(1)$ we get 

$0=3+\dfrac{(n-1)}{2}[2\times 2+(n-1-1)\times 2)]$

$T _{n}=3+(n-1)(n)$

$=3+n^{2}-n=n^{2}-n+3$

$S _{n}=\displaystyle\sum _{k=1}^{n}k^{2}=\sum _{k=1}^{n}k+3\sum _{k=1}^{n}1$

$=\dfrac{n(n+1)(2n+1)}{6}-\dfrac{n(n+1)}{2}+3n$

$=\dfrac{n(n+1)(2n+1)-3n(n+1)+18n}{6}$

$=\dfrac{n}{6}[2n^{2}+3n-3n-2+18]=\dfrac{n}{6}[2n^{2}+16]$

$=\dfrac{n}{6}\times 2[n^{2}+8]=\dfrac{n}{3}[n^{2}+8]$

Hence $S _{n}=\dfrac{n}{3}(n^{2}+8)$