activity 3: find the sum:
a. the first 30 integers
b. the first 50 integers
c. the first 100 integers
d. the first 782 integers

activity 4: given the following sequence: 4, 15, 26, …
a. what is the explicit and recursive formula for the sequence?
b. what is the sum of the first 5 terms in the sequence?

activity 5: given the following sequence: 82, 73, 64, …
a. what is the explicit and recursive formula for the sequence?
b. what is the sum of the first 5 terms in the sequence?

activity 6: given the following sequence: 5, 8, 11, …
a. what is the explicit and recursive formula for the sequence?
b. what is the sum of the terms ( a_7 - a_{23} ) in the sequence?

Question

activity 3: find the sum:
a. the first 30 integers
b. the first 50 integers
c. the first 100 integers
d. the first 782 integers

activity 4: given the following sequence: 4, 15, 26, …
a. what is the explicit and recursive formula for the sequence?
b. what is the sum of the first 5 terms in the sequence?

activity 5: given the following sequence: 82, 73, 64, …
a. what is the explicit and recursive formula for the sequence?
b. what is the sum of the first 5 terms in the sequence?

activity 6: given the following sequence: 5, 8, 11, …
a. what is the explicit and recursive formula for the sequence?
b. what is the sum of the terms ( a_7 - a_{23} ) in the sequence?

Accepted Answer

### Activity 3:
#### a. The first 30 integers
# Explanation:
## Step1: Identify the sequence
The first integers start from 1, so the sequence is $ 1, 2, 3, \dots, 30 $, which is an arithmetic sequence with $ a_1 = 1 $, $ d = 1 $, and $ n = 30 $.

## Step2: Use the sum formula for arithmetic series
The sum of the first $ n $ terms of an arithmetic series is $ S_n = \frac{n(a_1 + a_n)}{2} $. Here, $ a_n = 30 $, so:
$ S_{30} = \frac{30(1 + 30)}{2} $

## Step3: Calculate the sum
$ S_{30} = \frac{30 \times 31}{2} = 15 \times 31 = 465 $

# Answer:
$ 465 $

#### b. The first 50 integers
# Explanation:
## Step1: Identify the sequence
Sequence: $ 1, 2, 3, \dots, 50 $ (arithmetic, $ a_1 = 1 $, $ d = 1 $, $ n = 50 $).

## Step2: Apply the sum formula
$ S_n = \frac{n(a_1 + a_n)}{2} $, where $ a_n = 50 $:
$ S_{50} = \frac{50(1 + 50)}{2} $

## Step3: Compute
$ S_{50} = \frac{50 \times 51}{2} = 25 \times 51 = 1275 $

# Answer:
$ 1275 $

#### c. The first 100 integers
# Explanation:
## Step1: Sequence details
$ 1, 2, \dots, 100 $ (arithmetic, $ a_1 = 1 $, $ a_n = 100 $, $ n = 100 $).

## Step2: Sum formula
$ S_{100} = \frac{100(1 + 100)}{2} $

## Step3: Calculate
$ S_{100} = 50 \times 101 = 5050 $

# Answer:
$ 5050 $

#### d. The first 782 integers
# Explanation:
## Step1: Sequence info
$ 1, 2, \dots, 782 $ (arithmetic, $ a_1 = 1 $, $ a_n = 782 $, $ n = 782 $).

## Step2: Sum formula
$ S_{782} = \frac{782(1 + 782)}{2} $

## Step3: Compute
$ S_{782} = \frac{782 \times 783}{2} = 391 \times 783 $
$ 391 \times 783 = 391 \times (700 + 80 + 3) = 391 \times 700 + 391 \times 80 + 391 \times 3 $
$ = 273700 + 31280 + 1173 = 273700 + 31280 = 304980 + 1173 = 306153 $

# Answer:
$ 306153 $

### Activity 4: Sequence $ 4, 15, 26, \dots $
#### a. Explicit and recursive formulas
# Explanation:
## Step1: Identify the sequence type
Check the common difference: $ 15 - 4 = 11 $, $ 26 - 15 = 11 $. So it’s an arithmetic sequence with $ a_1 = 4 $, $ d = 11 $.

## Step2: Explicit formula
For an arithmetic sequence, $ a_n = a_1 + (n - 1)d $. Substituting $ a_1 = 4 $, $ d = 11 $:
$ a_n = 4 + (n - 1) \times 11 = 11n - 7 $

## Step3: Recursive formula
Recursive formula for arithmetic sequence: $ a_1 = 4 $, $ a_n = a_{n - 1} + 11 $ (for $ n \geq 2 $)

# Answer:
- Explicit: $ \boldsymbol{a_n = 11n - 7} $
- Recursive: $ \boldsymbol{a_1 = 4,\ a_n = a_{n - 1} + 11\ (n \geq 2)} $

#### b. Sum of the first 5 terms
# Explanation:
## Step1: Find the first 5 terms
Using $ a_n = 11n - 7 $:
- $ a_1 = 4 $
- $ a_2 = 11(2) - 7 = 15 $
- $ a_3 = 11(3) - 7 = 26 $
- $ a_4 = 11(4) - 7 = 37 $
- $ a_5 = 11(5) - 7 = 48 $

## Step2: Sum the terms
$ S_5 = 4 + 15 + 26 + 37 + 48 $

## Step3: Calculate
$ 4 + 15 = 19 $; $ 19 + 26 = 45 $; $ 45 + 37 = 82 $; $ 82 + 48 = 130 $

# Answer:
$ 130 $

### Activity 5: Sequence $ 82, 73, 64, \dots $
#### a. Explicit and recursive formulas
# Explanation:
## Step1: Sequence type
Common difference: $ 73 - 82 = -9 $, $ 64 - 73 = -9 $. Arithmetic sequence with $ a_1 = 82 $, $ d = -9 $.

## Step2: Explicit formula
$ a_n = a_1 + (n - 1)d = 82 + (n - 1)(-9) = 91 - 9n $

## Step3: Recursive formula
$ a_1 = 82 $, $ a_n = a_{n - 1} - 9 $ (for $ n \geq 2 $)

# Answer:
- Explicit: $ \boldsymbol{a_n = 91 - 9n} $
- Recursive: $ \boldsymbol{a_1 = 82,\ a_n = a_{n - 1} - 9\ (n \geq 2)} $

#### b. Sum of the first 5 terms
# Explanation:
## Step1: Find the first 5 terms
Using $ a_n = 91 - 9n $:
- $ a_1 = 82 $
- $ a_2 = 91 - 18 = 73 $
- $ a_3 = 91 - 27 = 64 $
- $ a_4 = 91 - 36 = 55 $
- $ a_5 = 91 - 45 = 46 $

## Step2: Sum the terms
$ S_5 = 82 + 73 + 64 + 55 + 46 $

## Step3: Calculate
$ 82 + 73 = 155 $; $ 155 + 64 = 219 $; $ 219 + 55 = 274 $; $ 274 + 46 = 320 $

# Answer:
$ 320 $

### Activity 6: Sequence $ 5, 8, 11, \dots $
#### a. Explicit and recursive formulas
# Explanation:
## Step1: Sequence type
Common difference: $ 8 - 5 = 3 $, $ 11 - 8 = 3 $. Arithmetic sequence with $ a_1 = 5 $, $ d = 3 $.

## Step2: Explicit formula
$ a_n = 5 + (n - 1) \times 3 = 3n + 2 $

## Step3: Recursive formula
$ a_1 = 5 $, $ a_n = a_{n - 1} + 3 $ (for $ n \geq 2 $)

# Answer:
- Explicit: $ \boldsymbol{a_n = 3n + 2} $
- Recursive: $ \boldsymbol{a_1 = 5,\ a_n = a_{n - 1} + 3\ (n \geq 2)} $

#### b. Sum of terms $ a_7 $ to $ a_{23} $
# Explanation:
## Step1: Find the number of terms
Number of terms $ k = 23 - 7 + 1 = 17 $

## Step2: Find $ a_7 $ and $ a_{23} $
Using $ a_n = 3n + 2 $:
- $ a_7 = 3(7) + 2 = 23 $
- $ a_{23} = 3(23) + 2 = 71 $

## Step3: Sum of arithmetic series from $ a_7 $ to $ a_{23} $
The sum of an arithmetic series is $ S_k = \frac{k(a_7 + a_{23})}{2} $. Substituting $ k = 17 $, $ a_7 = 23 $, $ a_{23} = 71 $:
$ S_{17} = \frac{17(23 + 71)}{2} = \frac{17 \times 94}{2} = 17 \times 47 = 799 $

# Answer:
$ 799 $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Activity 3:

a. The first 30 integers

Step1: Identify the sequence

Step2: Use the sum formula for arithmetic series

Step3: Calculate the sum

Step1: Identify the sequence

Step2: Apply the sum formula

Step3: Compute

Step1: Sequence details

Step2: Sum formula

Step3: Calculate

Answer:

b. The first 50 integers