UPSKILL MATH PLUS

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Learn moreA sequence is a function defined on the set of natural numbers \(\mathbb{N}\). A sequence is a function of form $f:\mathrm{\mathbb{N}}\to \mathrm{\mathbb{R}}$, 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 with 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 for a sequence is defined as**$f(n)={a}_{n}=\frac{3n+4}{n+6}$.

**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**:

$f(n)={a}_{n}=\frac{3n+4}{n+6}$, where \(n = 1, 2, 3, 4, ...\)

${a}_{1}=\frac{3(1)+4}{1+6}$

$=\frac{3+4}{7}$

${a}_{1}=\frac{7}{7}$

${a}_{2}=\frac{3(2)+4}{2+6}$

$=\frac{6+4}{8}$

${a}_{2}=\frac{10}{8}$

${a}_{3}=\frac{3(3)+4}{3+6}$

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

${a}_{3}=\frac{13}{9}$

**Therefore**,

**the first three terms are**$\frac{7}{7}$, $\frac{10}{8}$

**and**$\frac{13}{9}$.