5. Observe the following sequences and answer the questions. $$ \begin{array}{lll} ext { (a) } 4,7,12,19, \ldots & ext { (b) } 1,7,17,31, \ldots . & ext { (c) } 2,1,-2,-7 \ ext { (i) Add next three terms in each sequence. }\end{array} $$ (ii) Find the general term of the given sequences. 6tudy the following sequences and answer of the questions. $$ \begin{array}{lll} ext { (a) } 1,2,3,4, \ldots & ext { (b) } 8,3,-2,-7, \ldots & ext { (c) } 5,11,19 \ ext { (d) } \frac{3}{5}, \frac{4}{7}, \frac{6}{9}, \frac{9}{11}, \ldots & ext { (e) } 0.28,0.35,0.42,0.49, \ldots & ext { (f) } \frac{3}{2048}, \frac{1}{10}\end{array} $$ $$ \begin{array}{ll} ext { (i) Find the } \mathrm{n}^{ ext {th }} ext { term of the given sequences. } & \end{array} $$

Question

5. Observe the following sequences and answer the questions. $$ \begin{array}{lll}	ext { (a) } 4,7,12,19, \ldots & 	ext { (b) } 1,7,17,31, \ldots . & 	ext { (c) } 2,1,-2,-7 \ 	ext { (i) Add next three terms in each sequence. }\end{array} $$ (ii) Find the general term of the given sequences. 6tudy the following sequences and answer of the questions. $$ \begin{array}{lll}	ext { (a) } 1,2,3,4, \ldots & 	ext { (b) } 8,3,-2,-7, \ldots & 	ext { (c) } 5,11,19 \ 	ext { (d) } \frac{3}{5}, \frac{4}{7}, \frac{6}{9}, \frac{9}{11}, \ldots & 	ext { (e) } 0.28,0.35,0.42,0.49, \ldots & 	ext { (f) } \frac{3}{2048}, \frac{1}{10}\end{array} $$ $$ \begin{array}{ll}	ext { (i) Find the } \mathrm{n}^{	ext {th }} 	ext { term of the given sequences. } & \end{array} $$

UpStudy AI · Accepted Answer

Let's solve the problems step by step.

### Problem 5

#### Sequence (a): $$ 4, 7, 12, 19, \ldots $$

1. **Finding the next three terms:**
   - The differences between the terms are:
     - $$ 7 - 4 = 3 $$
     - $$ 12 - 7 = 5 $$
     - $$ 19 - 12 = 7 $$
   - The differences are increasing by 2: $$ 3, 5, 7, \ldots $$
   - The next difference will be $$ 9 $$:
     - Next term: $$ 19 + 9 = 28 $$
   - The next difference will be $$ 11 $$:
     - Next term: $$ 28 + 11 = 39 $$
   - The next difference will be $$ 13 $$:
     - Next term: $$ 39 + 13 = 52 $$

So, the next three terms are $$ 28, 39, 52 $$.

2. **Finding the general term:**
   - The sequence can be expressed as:
     $$
     a_n = a_{n-1} + (2n + 1)
     $$
   - This is a quadratic sequence. We can find the general term by observing the pattern:
     $$
     a_n = n^2 + 3n + 1
     $$

#### Sequence (b): $$ 1, 7, 17, 31, \ldots $$

1. **Finding the next three terms:**
   - The differences between the terms are:
     - $$ 7 - 1 = 6 $$
     - $$ 17 - 7 = 10 $$
     - $$ 31 - 17 = 14 $$
   - The differences are increasing by 4: $$ 6, 10, 14, \ldots $$
   - The next difference will be $$ 18 $$:
     - Next term: $$ 31 + 18 = 49 $$
   - The next difference will be $$ 22 $$:
     - Next term: $$ 49 + 22 = 71 $$
   - The next difference will be $$ 26 $$:
     - Next term: $$ 71 + 26 = 97 $$

So, the next three terms are $$ 49, 71, 97 $$.

2. **Finding the general term:**
   - The sequence can be expressed as:
     $$
     a_n = 2n^2 + 4n - 5
     $$

#### Sequence (c): $$ 2, 1, -2, -7 $$

1. **Finding the next three terms:**
   - The differences between the terms are:
     - $$ 1 - 2 = -1 $$
     - $$ -2 - 1 = -3 $$
     - $$ -7 - (-2) = -5 $$
   - The differences are increasing by -2: $$ -1, -3, -5, \ldots $$
   - The next difference will be $$ -7 $$:
     - Next term: $$ -7 - 7 = -14 $$
   - The next difference will be $$ -9 $$:
     - Next term: $$ -14 - 9 = -23 $$
   - The next difference will be $$ -11 $$:
     - Next term: $$ -23 - 11 = -34 $$

So, the next three terms are $$ -14, -23, -34 $$.

2. **Finding the general term:**
   - The sequence can be expressed as:
     $$
     a_n = -n^2 + n + 2
     $$

### Problem 6

#### Sequence (a): $$ 1, 2, 3, 4, \ldots $$

1. **Finding the $$ n^{th} $$ term:**
   - The general term is:
     $$
     a_n = n
     $$

#### Sequence (b): $$ 8, 3, -2, -7, \ldots $$

1. **Finding the $$ n^{th} $$ term:**
   - The differences between the terms are:
     - $$ 3 - 8 = -5 $$
     - $$ -2 - 3 = -5 $$
     - $$ -7 - (-2) = -5 $$
   - The sequence is linear with a common difference of $$ -5 $$:
     $$
     a_n = -5n + 13
     $$

#### Sequence (c): $$ 5, 11, 19 $$

1. **Finding the $$ n^{th} $$ term:**
   - The differences between the terms are:
     - $$ 11 - 5 = 6 $$
     - $$ 19 - 11 = 8 $$
   - The differences are increasing by 2:
     $$
     a_n = n^2 + 4n + 1
     $$

#### Sequence (d): $$ \frac{3}{5}, \frac{4}{7}, \frac{6}{9}, \frac{9}{11}, \ldots $$

1. **Finding the $$ n^{th} $$ term:**
   - The numerators are $$ 3, 4, 6, 9 $$ (which can be expressed as $$ n^2 + n + 1 $$).
   - The denominators are $$ 5, 7, 9, 11 $$ (which can be expressed as $$ 2n + 3 $$):
     $$
     a_n = \frac{n^2 + n + 1}{2n + 3}
     $$

#### Sequence (e): $$ 0.28, 0.35, 0.42, 0.49, \ldots $$

1. **Finding the $$ n^{th} $$ term:**
   - The differences between the terms are $$ 0.07 $$:
     $$
     a_n = 0.28 + 0.07(n - 1) = 0.21 + 0.07n
     $$

#### Sequence (f): $$ \frac{3}{2048}, \frac{1}{10} $$

1. **Finding the $$ n^{th} $$ term:**
   - This sequence appears to be decreasing rapidly. We can express it as:
     $$
     a_n = \frac{3}{2^{n+7}}
     $$

### Summary of Results

- **Problem 5:**
  - (a) Next terms: $$ 28, 39, 52 $$; General term: $$ n^2 + 3n + 1 $$
  - (b) Next terms: $$ 49, 71, 97 $$; General term: $$ 2n^2 + 4n - 5 $$
  - (c) Next terms: $$ -14,
- **Problem 5:** - (a) Next three terms: 28, 39, 52; General term: $$ n^2 + 3n + 1 $$ - (b) Next three terms: 49, 71, 97; General term: $$ 2n^2 + 4n - 5 $$ - (c) Next three terms: -14, -23, -34; General term: $$ -n^2 + n + 2 $$ - **Problem 6:** - (a) $$ a_n = n $$ - (b) $$ a_n = -5n + 13 $$ - (c) $$ a_n = n^2 + 4n + 1 $$ - (d) $$ a_n = \frac{n^2 + n + 1}{2n + 3} $$ - (e) $$ a_n = 0.21 + 0.07n $$ - (f) $$ a_n = \frac{3}{2^{n+7}} $$

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