$3$ . Escriba una expresión que represente el término $n$ -ésimo de las siguientes sucesiones. $\newline$ (a) $0,3,6,9,12 \ldots$ $\newline$ (b) $\frac{3}{4}, \frac{9}{16}, \frac{27}{64}, \frac{81}{256}, \ldots$ $\newline$ (c) $2, \frac{1}{3}, 4, \frac{1}{5}, 6, \frac{1}{7}, 8, \frac{1}{9}, \ldots$ $\newline$ (d) $-3,5,-7,9,-11, \ldots$

Full solution

3

. Escriba una expresión que represente el término

n

-ésimo de las siguientes sucesiones.

\newline

(a)

0,3,6,9,12 \ldots

\newline

(b)

\frac{3}{4}, \frac{9}{16}, \frac{27}{64}, \frac{81}{256}, \ldots

\newline

(c)

2, \frac{1}{3}, 4, \frac{1}{5}, 6, \frac{1}{7}, 8, \frac{1}{9}, \ldots

\newline

(d)

-3,5,-7,9,-11, \ldots

Arithmetic Sequence Explanation: (a) Sequence: $0$ , $3$ , $6$ , $9$ , $12$ , ... $\newline$ Reasoning: This is an arithmetic sequence with a common difference of $3$ . $\newline$ Calculation: The n-th term of an arithmetic sequence is given by $a_n = a_1 + (n-1)d$ . $\newline$ Here, $a_1 = 0$ and $d = 3$ . $\newline$ So, $a_n = 0 + (n-1) \cdot 3 = 3(n-1)$ .
Geometric Sequence Explanation: (b) Sequence: $\frac{3}{4}, \frac{9}{16}, \frac{27}{64}, \frac{81}{256}, \dots$ $\newline$ Reasoning: This is a geometric sequence with a common ratio of $\frac{3}{4}$ . $\newline$ Calculation: The n-th term of a geometric sequence is given by $a_n = a_1 \cdot r^{(n-1)}$ . $\newline$ Here, $a_1 = \frac{3}{4}$ and $r = \frac{3}{4}$ . $\newline$ So, $a_n = \left( \frac{3}{4} \right) \cdot \left( \frac{3}{4} \right)^{(n-1)} = \left( \frac{3}{4} \right)^n$ .
Integer and Fraction Sequence Explanation: (c) Sequence: $2$ , $\frac{1}{3}$ , $4$ , $\frac{1}{5}$ , $6$ , $\frac{1}{7}$ , $8$ , $\frac{1}{9}$ , ... $\newline$ Reasoning: This sequence alternates between integers and fractions. $\newline$ Calculation: For even n (integers): $a_{2k} = 2k$ . $\newline$ For odd n (fractions): $a_{2k-1} = \frac{1}{2k+1}$ .
Alternating Sequence Explanation: (d) Sequence: $-3$ , $5$ , $-7$ , $9$ , $-11$ , ... $\newline$ Reasoning: This is an alternating sequence with a pattern in the absolute values. $\newline$ Calculation: The n-th term can be represented as $a_n = (-1)^{n+1} \cdot (2n-1)$ .