Tag: series

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Multiple choice maths numbers and sequences series introduction to series introduction to sequences and series

A series is:

  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
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

Multiple choice maths numbers and sequences series introduction to series introduction to sequences and series

Series can be defined as:

  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
Reveal answer Fill a bubble to check yourself
D Correct answer
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
Multiple choice maths numbers and sequences series introduction to series introduction to sequences and series

A divergent 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
Reveal answer Fill a bubble to check yourself
C Correct answer
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.
Multiple choice maths numbers and sequences series introduction to series introduction to sequences and series

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

  1. $5$
  2. $13$
  3. $43$
  4. $176$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The pattern is: multiply by position, then subtract consecutive primes starting from 2. 3*1-? = 3, 5*2-2 = 8, 8*3-3 = 21 (but shows 13), 13*4-5 = 47 (but shows 43), 43*5-7 = 208 (but shows 176), 176*6-11 = 1045 (but shows 891). The pattern is complex; 176 fits as 43*4+8 = 180 or similar variation. Actually, checking the recursive pattern: 3*1+2=5, 5*2+3=13, 13*3+4=43 (NOT 176, should be 43), 43*4+5=177 (NOT 176), so 176 is WRONG - should be 43.

Multiple choice maths numbers and sequences series introduction to series introduction to sequences and series

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

  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 } $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The series is a sum of geometric series. The n-th term is (x^(n+1) - y^(n+1)) / (x-y). Summing this from n=1 to infinity gives the sum of two infinite geometric series: x/(1-x) + y/(1-y) is incorrect; the correct sum is (x+y-xy)/((1-x)(1-y)).

Multiple choice maths numbers and sequences series introduction to series introduction to sequences and series

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

  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
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $S _{n}=\dfrac{n}{3}(n^{2}+8)$