UPSKILL MATH PLUS

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### Theory:

A sequence is a function defined on the set of natural numbers $$\mathbb{N}$$. A sequence is a function of form $f:\mathrm{ℕ}\to \mathrm{ℝ}$, where $$\mathbb{R}$$ is the set of all real numbers.
If the sequence is of the form  $$a_1$$, $$a_2$$, $$a_3$$, $$a_4$$, ... then we can associate the function to the sequence $$a_1$$, $$a_2$$, $$a_3$$, $$a_4$$, ... by $$f (n) = a_n$$, $$n = 1, 2, 3,$$ …

Let us solve a problem of sequence with a function to understand this concept.
Example:
The general term of a sequence is defined as $f\left(n\right)={a}_{n}=\frac{2n+4}{n+7}$. Find the first three terms $$a_1$$, $$a_2$$, ${a}_{3}$.

Let us substitute the natural number $$n = 1, 2, 3, 4, ...$$ in the given equation.

The first three terms are:

$f\left(n\right)={a}_{n}=\frac{2n+4}{n+7}$, where $$n = 1, 2, 3, 4, ...$$

${a}_{1}=\frac{2\left(1\right)+4}{1+7}$

$=\frac{2+4}{8}$

${a}_{1}=\frac{6}{8}$

${a}_{2}=\frac{2\left(2\right)+4}{2+7}$

$=\frac{4+4}{9}$

${a}_{2}=\frac{8}{9}$

${a}_{3}=\frac{2\left(3\right)+4}{3+7}$

$=\frac{6+4}{10}$

${a}_{3}=\frac{10}{10}$

Therefore, the first three terms are $\frac{6}{8}$, $\frac{8}{9}$ and $\frac{10}{10}$.