3.1 Given the arithmetic series: $$ 18+24+30+\ldots+300 $$ 3.1.1 Determine the number of terms in this series 3.1.2 Calculate the sum of this series 3.1.3 Calculate the sum of all the whole numbers up to and including 300 that are NOT divisible by 6

Question

UpStudy AI · Accepted Answer

**3.1.1 Determine the number of terms in the series**

The arithmetic series is  
$$
18,\,24,\,30,\ldots,300
$$  
with first term $$a_1 = 18$$, common difference $$d = 6$$, and last term $$a_n = 300$$.

We use the formula for the $$n$$th term of an arithmetic sequence:
$$
a_n = a_1 + (n-1)d
$$
Plug in the values:
$$
300 = 18 + (n-1) \cdot 6
$$
Subtract $$18$$ from both sides:
$$
300 - 18 = 6(n-1)
$$
$$
282 = 6(n-1)
$$
Divide both sides by $$6$$:
$$
n-1 = \frac{282}{6} = 47
$$
Thus, 
$$
n = 47 + 1 = 48
$$

**3.1.2 Calculate the sum of the series**

The sum $$S_n$$ of an arithmetic series is given by:
$$
S_n = \frac{n}{2}(a_1 + a_n)
$$
Substitute $$n = 48$$, $$a_1 = 18$$, and $$a_n = 300$$:
$$
S_{48} = \frac{48}{2}(18 + 300)
$$
Simplify:
$$
S_{48} = 24 \times 318 = 7632
$$

**3.1.3 Calculate the sum of all whole numbers up to and including 300 that are NOT divisible by 6**

1. **Sum of all whole numbers from 1 to 300:**

The sum of the first $$N$$ natural numbers is:
   $$
   S = \frac{N(N+1)}{2}
   $$
   For $$N = 300$$:
   $$
   S = \frac{300 \times 301}{2} = 150 \times 301 = 45150
   $$

2. **Sum of all multiples of 6 up to 300:**

Multiples of 6 form an arithmetic series:
   $$
   6,\,12,\,18,\ldots,300
   $$
   Here, the first term is $$6$$ and the last term is $$300$$. The number of terms is given by:
   $$
   n = \frac{300}{6} = 50
   $$
   The sum of this series is:
   $$
   S_6 = \frac{n}{2}(6 + 300) = \frac{50}{2}(306) = 25 \times 306 = 7650
   $$

3. **Sum of numbers NOT divisible by 6:**

Subtract the sum of the multiples of 6 from the total sum:
   $$
   S_{\text{not 6}} = 45150 - 7650 = 37500
   $$

Thus, the answers are:

- **3.1.1** Number of terms: $$48$$
- **3.1.2** Sum of the series: $$7632$$
- **3.1.3** Sum of whole numbers up to 300 not divisible by 6: $$37500$$
**3.1.1** There are 48 terms in the series. **3.1.2** The sum of the series is 7,632. **3.1.3** The sum of all whole numbers up to and including 300 that are not divisible by 6 is 37,500.

3.1 Given the arithmetic series:
3.1.1 Determine the number of terms in this series
3.1.2 Calculate the sum of this series
3.1.3 Calculate the sum of all the whole numbers up to and including 300 that are
NOT divisible by 6

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Arithmetic Questions

3.1 Given the arithmetic series: 3.1.1 Determine the number of terms in this series 3.1.2 Calculate the sum of this series 3.1.3 Calculate the sum of all the whole numbers up to and including 300 that are NOT divisible by 6