ybghalan
Answered

Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Discover comprehensive answers to your questions from knowledgeable professionals on our user-friendly platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

2. Find the general terms [tex]\([t_n]\)[/tex] of the following sequences:

(a) [tex]\(4, 6, 8, 10, \ldots\)[/tex]

(b) [tex]\(7, 11, 15, 19, 23, \ldots\)[/tex]

(c) [tex]\(2, 6, 10, 14, 18, \ldots\)[/tex]

(d) [tex]\(25, 22, 19, 16, \ldots\)[/tex]

(e) [tex]\(\frac{1}{3}, \frac{4}{5}, \frac{7}{7}, \frac{10}{9}, \ldots\)[/tex]

(f) [tex]\(\frac{2}{7}, \frac{5}{8}, \frac{8}{9}, \frac{11}{10}, \ldots\)[/tex]

(g) [tex]\(40, 38, 36, 34, \ldots\)[/tex]

(h) [tex]\(\frac{2}{5}, \frac{4}{8}, \frac{6}{11}, \frac{8}{14}, \ldots\)[/tex]

Sagot :

GinnyAnswer
Let's find the general terms [tex]\( t_n \)[/tex] of the given sequences one by one.

### (a) [tex]\( 4, 6, 8, 10, \ldots \)[/tex]

This is an arithmetic sequence with a common difference [tex]\( d = 6 - 4 = 2 \)[/tex].

The general term for an arithmetic sequence is given by:
[tex]\[ t_n = a + (n - 1) d \][/tex]

Here, [tex]\( a = 4 \)[/tex] and [tex]\( d = 2 \)[/tex].

So,
[tex]\[ t_n = 4 + (n - 1) \cdot 2 = 2n + 2 \][/tex]

### (b) [tex]\( 7, 11, 15, 19, 23, \ldots \)[/tex]

This is an arithmetic sequence with a common difference [tex]\( d = 11 - 7 = 4 \)[/tex].

Here, [tex]\( a = 7 \)[/tex] and [tex]\( d = 4 \)[/tex].

So,
[tex]\[ t_n = 7 + (n - 1) \cdot 4 = 4n + 3 \][/tex]

### (c) [tex]\( 2, 6, 10, 14, 18, \ldots \)[/tex]

This is an arithmetic sequence with a common difference [tex]\( d = 6 - 2 = 4 \)[/tex].

Here, [tex]\( a = 2 \)[/tex] and [tex]\( d = 4 \)[/tex].

So,
[tex]\[ t_n = 2 + (n - 1) \cdot 4 = 4n - 2 \][/tex]

### (d) [tex]\( 25, 22, 19, 16, \ldots \)[/tex]

This is an arithmetic sequence with a common difference [tex]\( d = 22 - 25 = -3 \)[/tex].

Here, [tex]\( a = 25 \)[/tex] and [tex]\( d = -3 \)[/tex].

So,
[tex]\[ t_n = 25 + (n - 1) \cdot (-3) = 28 - 3n \][/tex]

### (e) [tex]\( \frac{1}{3}, \frac{4}{5}, \frac{7}{7}, \frac{10}{9}, \ldots \)[/tex]

Let's express these ratios in terms of a general formula. The general term here is more complex.

The general term can be provided as:
[tex]\[ t_n = \frac{8406719304424927 n}{18014398509481984} - \frac{1200959900632133}{9007199254740992} \][/tex]

### (f) [tex]\( \frac{2}{7}, \frac{5}{8}, \frac{8}{9}, \frac{11}{10}, \ldots \)[/tex]

Similar to (e), this is another sequence with a fractional relationship.

The general term can be expressed as:
[tex]\[ t_n = \frac{3056014032858551 n}{9007199254740992} - \frac{241264265751991}{4503599627370496} \][/tex]

### (g) [tex]\( 40, 38, 36, 34, \ldots \)[/tex]

This is an arithmetic sequence with a common difference [tex]\( d = 38 - 40 = -2 \)[/tex].

Here, [tex]\( a = 40 \)[/tex] and [tex]\( d = -2 \)[/tex].

So,
[tex]\[ t_n = 40 + (n - 1) \cdot (-2) = 42 - 2n \][/tex]

### (h) [tex]\( \frac{2}{5}, \frac{4}{8}, \frac{6}{11}, \frac{8}{14}, \ldots \)[/tex]

Like the previous fractional sequences, this one also adheres to a specific ratio pattern.

The general term for this sequence is:
[tex]\[ t_n = \frac{900719925474099n}{9007199254740992} + \frac{1351079888211149}{4503599627370496} \][/tex]

These are the general terms for each of the given sequences.
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.