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: