Sequence as a function — lesson. Mathematics State Board, Class 10.

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A sequence is a function defined on the set of natural numbers

$\mathbb{N}$ . A sequence is a function of form

$f : ℕ \to ℝ$ , 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

$f (n) = a_{n} = \frac{3 n + 4}{n + 6}$ . 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:

$f (n) = a_{n} = \frac{3 n + 4}{n + 6}$ , 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

$\frac{13}{9}$ .

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2. Sequence as a function

Theory: