ybghalan
Answered

At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Experience the ease of finding reliable answers to your questions from a vast community of knowledgeable experts. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

3. From the given general terms, find the first 5 terms and present them in the form of a sequence when [tex][tex]$n$[/tex][/tex] represents a natural number:

(a) [tex]t_n = 2n + 4[/tex]

(b) [tex]t_n = 3n - 1[/tex]

(c) [tex]t_n = 3^n[/tex]

(d) [tex]t_n = n^2 - 1[/tex]

(e) [tex]t_n = (-1)^n \cdot n^2[/tex]

(f) [tex]t_n = n^2 + 2n + 3[/tex]

(g) [tex]t_n = 3n^2 - 5[/tex]

Sagot :

GinnyAnswer
Certainly! Let's find the first five terms for each given sequence when [tex]\( n \)[/tex] represents the natural numbers:

### (a) [tex]\( t_n = 2n + 4 \)[/tex]

For [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 2(1) + 4 = 6 \)[/tex]
2. [tex]\( t_2 = 2(2) + 4 = 8 \)[/tex]
3. [tex]\( t_3 = 2(3) + 4 = 10 \)[/tex]
4. [tex]\( t_4 = 2(4) + 4 = 12 \)[/tex]
5. [tex]\( t_5 = 2(5) + 4 = 14 \)[/tex]

So, the first five terms are 6, 8, 10, 12, 14.

### (b) [tex]\( t_n = 3n - 1 \)[/tex]

For [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 3(1) - 1 = 2 \)[/tex]
2. [tex]\( t_2 = 3(2) - 1 = 5 \)[/tex]
3. [tex]\( t_3 = 3(3) - 1 = 8 \)[/tex]
4. [tex]\( t_4 = 3(4) - 1 = 11 \)[/tex]
5. [tex]\( t_5 = 3(5) - 1 = 14 \)[/tex]

So, the first five terms are 2, 5, 8, 11, 14.

### (c) [tex]\( t_n = 3^n \)[/tex]

For [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 3^1 = 3 \)[/tex]
2. [tex]\( t_2 = 3^2 = 9 \)[/tex]
3. [tex]\( t_3 = 3^3 = 27 \)[/tex]
4. [tex]\( t_4 = 3^4 = 81 \)[/tex]
5. [tex]\( t_5 = 3^5 = 243 \)[/tex]

So, the first five terms are 3, 9, 27, 81, 243.

### (d) [tex]\( t_n = n^2 - 1 \)[/tex]

For [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 1^2 - 1 = 0 \)[/tex]
2. [tex]\( t_2 = 2^2 - 1 = 3 \)[/tex]
3. [tex]\( t_3 = 3^2 - 1 = 8 \)[/tex]
4. [tex]\( t_4 = 4^2 - 1 = 15 \)[/tex]
5. [tex]\( t_5 = 5^2 - 1 = 24 \)[/tex]

So, the first five terms are 0, 3, 8, 15, 24.

### (e) [tex]\( t_n = (-1)^n \cdot n^2 \)[/tex]

For [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = (-1)^1 \cdot 1^2 = -1 \)[/tex]
2. [tex]\( t_2 = (-1)^2 \cdot 2^2 = 4 \)[/tex]
3. [tex]\( t_3 = (-1)^3 \cdot 3^2 = -9 \)[/tex]
4. [tex]\( t_4 = (-1)^4 \cdot 4^2 = 16 \)[/tex]
5. [tex]\( t_5 = (-1)^5 \cdot 5^2 = -25 \)[/tex]

So, the first five terms are -1, 4, -9, 16, -25.

### (f) [tex]\( t_n = n^2 + 2n + 3 \)[/tex]

For [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 1^2 + 2(1) + 3 = 6 \)[/tex]
2. [tex]\( t_2 = 2^2 + 2(2) + 3 = 11 \)[/tex]
3. [tex]\( t_3 = 3^2 + 2(3) + 3 = 18 \)[/tex]
4. [tex]\( t_4 = 4^2 + 2(4) + 3 = 27 \)[/tex]
5. [tex]\( t_5 = 5^2 + 2(5) + 3 = 38 \)[/tex]

So, the first five terms are 6, 11, 18, 27, 38.

### (g) [tex]\( t_n = 3n^2 - 5 \)[/tex]

For [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 3(1^2) - 5 = -2 \)[/tex]
2. [tex]\( t_2 = 3(2^2) - 5 = 7 \)[/tex]
3. [tex]\( t_3 = 3(3^2) - 5 = 22 \)[/tex]
4. [tex]\( t_4 = 3(4^2) - 5 = 43 \)[/tex]
5. [tex]\( t_5 = 3(5^2) - 5 = 70 \)[/tex]

So, the first five terms are -2, 7, 22, 43, 70.

These are the sequences for each given general term.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.