Sequence And Series - Practice Questions & MCQ

Sequences:
A sequence is formed when terms are written in order such that they follow a particular pattern.

$
\begin{aligned}
& \text { E.g. } 1,2,3,4,5, \ldots . . \\
& 1,4,9,16, \ldots \ldots \\
& 1 / 3,1 / 4,1 / 5,1 / 6, \ldots \ldots
\end{aligned}
$
Finite and Infinite Sequences
A sequence may have an infinite number of terms or a finite number of terms.
If a sequence has three dots at the end, then it indicates the list never ends. It has an infinite number of terms. Then, the sequence is said to be an infinite sequence.
E.g., 2, 4, 6, 8, 10,

If the sequence has only a finite number of terms, then the sequence is called a finite sequence.
Eg, 2, 4, 6, 8

$
5,8,11,14, \ldots \ldots ., 65
$

$\mathrm{n}^{\text {th }}$ term
In sequences $n^{\text {th }}$ term is usually denoted by $a_n$ or $t_n$ or $T_n$
So, in sequence $2,4,6,8, \ldots$
$a_1=2, a_2=4, a_3=6$, and so on
We can write it in compact form as $\mathrm{a}_{\mathrm{n}}=2 \mathrm{n}$
$a_n$ or $n^{\text {th }}$ term is also called the General term of the sequence.
So a sequence can be written as $a_1, a_2, a_3$,
Conversely, if the general term of a sequence is given, we can find any term of that sequence.
Eg, If $T_n=2^n$, then fourth term can be obtained by putting $n=4$
So, $T_4=2^4=16$

Series:
If we add or subtract all the terms of a sequence we will get an expression, which is called a series. It is denoted by $\mathrm{S}_{\mathrm{n}}$. If the sequence is $a_1, a_2, a_3, \ldots \ldots, a_n$, then it's sum i.e. $a_1+a_2+a_3+\ldots \ldots \ldots+a_n$ is a series.

$
\mathrm{S}_{\mathrm{n}}=a_1+a_2+a_3+\ldots \ldots \ldots .+a_n=\sum_{\mathrm{r}=1}^{\mathrm{n}} a_r=\sum a_T
$
Then,

$
\begin{aligned}
\mathrm{S}_{\mathrm{n}}-\mathrm{S}_{\mathrm{n}-1}=\left(a_1+a_2+a_3\right. & \left.+\ldots \ldots \ldots .+a_{n-1}+a_n\right) \\
& -\left(a_1+a_2+a_3+\ldots \ldots \ldots .+a_{n-1}\right)
\end{aligned}
$
Thus, $\quad a_n=\mathrm{S}_{\mathrm{n}}-\mathrm{S}_{\mathrm{n}-1}$
This is the formula for finding the general term of a sequence if the sum of n terms is given.

Progression:

If the terms of a sequence follow some pattern that can be defined by an explicit formula in n, then the sequence is called a progression. 

Eg. 3, 9, 27, $81 \ldots$
Here, the general term can be written explicitly in terms of $n$, which is $3^n$