Mathematical Series and Sequences

When each term of a sequence is connected using a $+$ or a $-$ sign, then it is referred to as the series of numbers. For example, $2 + 5 + 8 + 11 + 15 + 18 + ....$ is a series of numbers. Thus the correct answer is '$a$'.

The series is $\text{x++/xxx+++/ xxxx++++/x}$. Thus, the letters are repeated twice, thrice and so on.

Adding all the terms in a sequence is called series.
Example: 1 + 2 + 3 + 4 is a series.

$1 + 2 + 3 + 4 + 5 +.....$ is a series.
As sum of all numbers in some set is called as a series.

A sum of an infinite sequence it is called a series.
Example: $2 + 4 + 6 + 8 +....$ is a series.

$1 + 2 + 3 + 4 + 5$ is a series.

The sum of first $5$ odd numbers is called series.
Example: $1 + 3 + 5 + 7 + 9 $ is a series.

Given series is $1 + 3 + 5 + 7 $ ____

Adding first $100$ terms in a sequence is called series.

A series is a sum of numbers.

Series is adding all the numbers or sum of all numbers.

A series is the sum of some set of terms of a sequence.
So, multiplying first $10$ odd numbers is not a series.

$\displaystyle \frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+....$ is a series.

Adding and constant difference between the terms is called series.
Example: $1 + 3 + 5 + 7 +... $
Here the common difference is $2$, adding the terms is called series.

A series is the sum of some set of terms of a sequence.

Given:  $\displaystyle\sum _{n=0}^{4} 2n$

$(A)$ $1,2,3,4,5,6,....$

Arithmetic progression, Geometric progression and Fibonacci pattern are all series.

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.

The pattern is: multiply by position, then subtract consecutive primes starting from 2. 31-? = 3, 52-2 = 8, 83-3 = 21 (but shows 13), 134-5 = 47 (but shows 43), 435-7 = 208 (but shows 176), 1766-11 = 1045 (but shows 891). The pattern is complex; 176 fits as 434+8 = 180 or similar variation. Actually, checking the recursive pattern: 31+2=5, 52+3=13, 133+4=43 (NOT 176, should be 43), 43*4+5=177 (NOT 176), so 176 is WRONG - should be 43.

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)).

The sequence $1,1,1,....$ is in A.P whose first term $a=1 ,d=1-1=0$

If a series consists only a finite number of terms it is called a Finite series.

$S _{75} = 2625$
$\Rightarrow \dfrac {75}{2}(2a + (75 - 1)d) = 2625$
$\Rightarrow 2(a + 37d) = 35\times 2$
$\Rightarrow a + 37d = 35$
$T _{38} = a + (38 - 1)d = a + 37 d$
$= 35$

Given:
 $\Rightarrow f(n+1)=\dfrac {2f(n)+1}{2}$
and $\Rightarrow  f(1)=2$
For  $n=1$,$ f(2)=\dfrac {2f(1)+1}{2}=\dfrac {5}{2}$

If an AP consist of $30$ terms, Then $x _{1} + x _{30} = x _{4} + x _{27} = x _{9} + x _{22} = x _{11} + x _{20}$
$\because x _{1} +x _{4} + x _{9} + x _{11} + x _{20} + x _{27} + x _{30} = 272$
$\Rightarrow (x _{1} + x _{30}) + (x _{4} + x _{27}) + (x _{9} + x _{22}) + (x _{11} + x _{26}) = 272$
$\Rightarrow 4(x _{1} + x _{30}) = 272$
$\Rightarrow x _{1} + x _{30} = \dfrac {272}{4} = 68$
$S _{30} = \dfrac {30}{2} (x _{1} + x _{30}) = 15\times 68 = 1020$

$S _{n} = 1^{3} + 2^{3} + ...... + n^{3} = \sum n^{3}$
$T _{n} = 1 + 2 + ..... + n = \sum n$
$S _{n} = \sum n^{3} = \left [\dfrac {n(n + 1)}{2}\right ]^{2}$
$\Rightarrow S _{n} = \left {\sum n\right }^{2} = T _{n}^{2}$

The sum is a geometric series: ( (1+x)^(n+1) - 1 ) / ( (1+x) - 1 ) = ( (1+x)^(n+1) - 1 ) / x. The sum of coefficients a_k is the value of the polynomial at x=1, which is ((1+1)^(n+1) - 1) / 1 = 2^(n+1) - 1. Option A is the standard result for this series.

When $n = 1$, $1(1+3)=4$ i.e. $1(1+3)$
When $n = 2$, $2(2+3)=10$ i.e. $2(2+3)$
When $n = 3$, $3(3+3)=18$ i.e. $3(3+3)$
When $n = 4$, $4(4+3)=28$ i.e. $4(4+3)$
So, $4, 10, 27, 28..$ is the function for the sequence is $n(n+3)$.

Differences are $13, 26, 39$
$\Rightarrow 4^{th} $ difference $= 52$,
$\Rightarrow$ required number $= 80 + 52 = 132$.

Given ${a} _{k+1} = {a} _{k}+3$ and ${a} _{1}=7$