1. Determine the $$ 10^{ ext {th }} $$ and $$ 21^{ ext {st }} $$ terms of each of the arithmetic sequences: (a) $$ 7 ; 10 ; 13 ; \ldots $$ (c) $$ 4+7 x ; 5+9 x ; 6+11 x ; \ldots \ldots $$. (e) $$ a=-2 ; T_{5}=18 $$ 2. In each case, the general term of an arithmetic sequence is given. (i) Determine the $$ 1^{ ext {st }} $$ term. (ii) Determine the common difference. (iii) Determine the $$ 21^{ ext {st }} $$ term. a) $$ T_{k}=2+3 k $$ (c) $$ T_{k}=2 k-1 $$ e) $$ T_{k}=6+4 k $$

Question

1. Determine the $$ 10^{	ext {th }} $$ and $$ 21^{	ext {st }} $$ terms of each of the arithmetic sequences: (a) $$ 7 ; 10 ; 13 ; \ldots $$ (c) $$ 4+7 x ; 5+9 x ; 6+11 x ; \ldots \ldots $$. (e) $$ a=-2 ; T_{5}=18 $$ 2. In each case, the general term of an arithmetic sequence is given. (i) Determine the $$ 1^{	ext {st }} $$ term. (ii) Determine the common difference. (iii) Determine the $$ 21^{	ext {st }} $$ term. a) $$ T_{k}=2+3 k $$ (c) $$ T_{k}=2 k-1 $$ e) $$ T_{k}=6+4 k $$

UpStudy AI · Accepted Answer

Let's solve the problems step by step.

### Problem 1: Determine the $$ 10^{\text{th}} $$ and $$ 21^{\text{st}} $$ terms of each arithmetic sequence.

#### (a) Sequence: $$ 7, 10, 13, \ldots $$

1. **Identify the first term and common difference:**
   - First term $$ a = 7 $$
   - Common difference $$ d = 10 - 7 = 3 $$

2. **General term formula:**
   $$
   T_n = a + (n-1)d
   $$

3. **Calculate $$ T_{10} $$:**
   $$
   T_{10} = 7 + (10-1) \cdot 3 = 7 + 27 = 34
   $$

4. **Calculate $$ T_{21} $$:**
   $$
   T_{21} = 7 + (21-1) \cdot 3 = 7 + 60 = 67
   $$

#### (c) Sequence: $$ 4 + 7x, 5 + 9x, 6 + 11x, \ldots $$

1. **Identify the first term and common difference:**
   - First term $$ a = 4 + 7x $$
   - Common difference $$ d = (5 + 9x) - (4 + 7x) = 1 + 2x $$

2. **General term formula:**
   $$
   T_n = a + (n-1)d
   $$

3. **Calculate $$ T_{10} $$:**
   $$
   T_{10} = (4 + 7x) + (10-1)(1 + 2x) = (4 + 7x) + 9(1 + 2x) = 4 + 7x + 9 + 18x = 13 + 25x
   $$

4. **Calculate $$ T_{21} $$:**
   $$
   T_{21} = (4 + 7x) + (21-1)(1 + 2x) = (4 + 7x) + 20(1 + 2x) = (4 + 7x) + 20 + 40x = 24 + 47x
   $$

#### (e) Sequence: $$ a = -2; T_5 = 18 $$

1. **Identify the first term and find the common difference:**
   - First term $$ a = -2 $$
   - Use the general term formula to find $$ d $$:
   $$
   T_5 = a + 4d = 18 \implies -2 + 4d = 18 \implies 4d = 20 \implies d = 5
   $$

2. **General term formula:**
   $$
   T_n = -2 + (n-1) \cdot 5
   $$

3. **Calculate $$ T_{10} $$:**
   $$
   T_{10} = -2 + (10-1) \cdot 5 = -2 + 45 = 43
   $$

4. **Calculate $$ T_{21} $$:**
   $$
   T_{21} = -2 + (21-1) \cdot 5 = -2 + 100 = 98
   $$

### Summary of Problem 1 Results:
- (a) $$ T_{10} = 34, T_{21} = 67 $$
- (c) $$ T_{10} = 13 + 25x, T_{21} = 24 + 47x $$
- (e) $$ T_{10} = 43, T_{21} = 98 $$

---

### Problem 2: Determine the first term, common difference, and $$ 21^{\text{st}} $$ term for each arithmetic sequence.

#### (a) General term: $$ T_k = 2 + 3k $$

1. **Determine the $$ 1^{\text{st}} $$ term:**
   $$
   T_1 = 2 + 3 \cdot 1 = 5
   $$

2. **Determine the common difference:**
   $$
   d = T_2 - T_1 = (2 + 3 \cdot 2) - (2 + 3 \cdot 1) = (2 + 6) - (2 + 3) = 6 - 5 = 3
   $$

3. **Determine the $$ 21^{\text{st}} $$ term:**
   $$
   T_{21} = 2 + 3 \cdot 21 = 2 + 63 = 65
   $$

#### (c) General term: $$ T_k = 2k - 1 $$

1. **Determine the $$ 1^{\text{st}} $$ term:**
   $$
   T_1 = 2 \cdot 1 - 1 = 1
   $$

2. **Determine the common difference:**
   $$
   d = T_2 - T_1 = (2 \cdot 2 - 1) - (2 \cdot 1 - 1) = (4 - 1) - (2 - 1) = 3 - 1 = 2
   $$

3. **Determine the $$ 21^{\text{st}} $$ term:**
   $$
   T_{21} = 2 \cdot 21 - 1 = 42 - 1 = 41
   $$

#### (e) General term: $$ T_k = 6 + 4k $$

1. **Determine the $$ 1^{\text{st}} $$ term:**
   $$
   T_1 = 6 + 4 \cdot 1 = 10
   $$

2. **Determine the common difference:**
   $$
   d = T_2 - T_1 = (6 + 4 \cdot 2) - (6 + 4 \cdot 1) = (6 + 8) - (6 + 4) = 14 - 10 = 4
   $$

3. **Determine the $$ 21^{\text{st}} $$ term:**
   $$
   T_{21} = 6 + 4 \cdot 21 = 6 + 84 = 90
   $$

### Summary of Problem 2 Results:
- (a) $$ T_1 = 5, d = 3, T_{21} = 65 $$
- (c) $$ T_1 = 1, d = 2, T_{21} = 41 $$
- (e) $$ T_1 = 10, d = 4, T_{21} = 
- (a) First term: 5, common difference: 3, 21st term: 65 - (c) First term: 1, common difference: 2, 21st term: 41 - (e) First term: 10, common difference: 4, 21st term: 90

Determine the and terms of each of the arithmetic sequences:
(a)
© .
(e)

In each case, the general term of an arithmetic sequence is given.
(i) Determine the term.
(ii) Determine the common difference.
(iii) Determine the term.
a)
©
e)

Upstudy AI Solution

Answer

Solution

Problem 1: Determine the and terms of each arithmetic sequence.

(a) Sequence:

© Sequence:

(e) Sequence:

Summary of Problem 1 Results:

Problem 2: Determine the first term, common difference, and term for each arithmetic sequence.

(a) General term:

© General term:

(e) General term:

Summary of Problem 2 Results:

The Deep Dive

Related Questions

Latest Algebra Questions

Determine the and terms of each of the arithmetic sequences: (a) © . (e) In each case, the general term of an arithmetic sequence is given. (i) Determine the term. (ii) Determine the common difference. (iii) Determine the term. a) © e)