A Sequence is a succession of terms arranged in some definite order such that the terms are related to one another according to a well-defined rule.

10, 17, 24, 31, 38, ........

-14, -12, -10, -8, .........

**finite sequences**

**infinite sequences**

**If the nth term of a sequence is $2+4^n$. Find the first three terms of the sequence.**

__Example 1__**Solution:**

$T_n=2+4^n$

$T_1=2+4^1$

$T_1=2+4$

$T_1=6$

$T_2=2+16$

$T_2=18$

__Example 2__**Solution:**

__Example 3__**Solution:**

__Representation of sequence using factorial notation.__The formula for some sequence contains products or successive positive numbers which can be expressed in factorial notation.

A factorial notation is the product of all the positive numbers preceding or equivalent to $n$.

As an example,

$6!=6 \times 5 \times 4 \times 3 \times 2 \times 1=720$

An example of a sequence formula containing factorial is

$a_n=(n+2)!$

The 3rd term of the sequence above can be found by substituting 3 for n

$a_3=(3+2)!=5!$

$5 × 4 ×3 ×2 ×1=120$

__Example 4__

Write the first three terms of the sequence $a_n=\frac{3n}{(n+1)!}$

**Solution: **

Remember that $a_n=\frac{3n}{(n+1)!}$, hence

$a_1=\frac{3(1)}{((1)+1)!}$

$a_1=\frac{3}{2 ×1}$

$a_1=\frac{3}{2}$

$a_2=\frac{3(2)}{((2)+1)!}$

$a_2=\frac{6}{3 ×2 ×1}$

$a_2=\frac{6}{6}=1$

$a_3=\frac{3(3)}{((3)+1)!}$

$a_3=\frac{9}{4 × 3 ×2 ×1}$

$a_3=\frac{9}{24}=\frac{3}{8}$

Hence, the first, second and third terms of the sequence are $\frac{3}{2}$, 1, $\frac{3}{8}$ respectively

__Example 5__

Write the first three terms of the sequence $a_n=\frac{(n+2)!}{3n}$

**Solution:**

Remember that $a_n=\frac{(n+2)!}{3n}$

$a_1=\frac{(1+2)!}{3(1)}$

$a_1=\frac{3!}{3}$

$a_1=\frac{3 ×2 ×1}{3}$

$a_1=2$

$a_2=\frac{(2+2)!}{3(2)}$

$a_2=\frac{4!}{6}$

$a_2=\frac{4 × 3 ×2 ×1}{6}$

$a_2=4$

$a_3=\frac{(3+2)!}{3(3)}$

$a_3=\frac{5!}{9}$

$a_3=\frac{5 × 4 × 3 ×2 ×1}{9}$

$a_3=\frac{40}{3}$

The first terms of the sequence are 2, 4, $\frac{40}{3}$

__Series__

*The addition of the terms of a sequence is called a series. The series is usually denoted by $S_n$.*

__Example 6__**Solution:**

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