If n is always a natural number, how many terms in the following sequence are integers?
`210, 84, (105)/2,…,(420)/(3n-1)`

Options

A. A)3
B. B)5
C. C)6
D. D)none of these

Answer

To determine how many terms in the sequence \( 210, 84, \frac{105}{2}, \ldots, \frac{420}{3n-1} \) are integers, we will follow these steps:

### Step 1: Identify the general term in the sequence
The general term of the sequence is given by \( \frac{420}{3n - 1} \).

### Step 2: Determine the conditions for \( \frac{420}{3n - 1} \) to be an integer
For \( \frac{420}{3n - 1} \) to be an integer, \( 3n - 1 \) must be a divisor of 420. This means we need to find all values of \( n \) such that \( 3n - 1 \) is a factor of 420.

### Step 3: Find the factors of 420
First, we will find the factors of 420. The prime factorization of 420 is:
\[ 420 = 2^2 \times 3^1 \times 5^1 \times 7^1 \]
Using the prime factorization, we can find the total number of factors:
\[ (2+1)(1+1)(1+1)(1+1) = 3 \times 2 \times 2 \times 2 = 24 \]
So, 420 has 24 factors.

### Step 4: Identify which factors can be expressed as \( 3n - 1 \)
We need to express each factor \( d \) of 420 in the form \( 3n - 1 \). Rearranging gives us:
\[ 3n = d + 1 \implies n = \frac{d + 1}{3} \]
For \( n \) to be a natural number, \( d + 1 \) must be divisible by 3.

### Step 5: List the factors of 420 and check divisibility
The factors of 420 are:
1. 1
2. 2
3. 3
4. 4
5. 5
6. 6
7. 7
8. 10
9. 12
10. 14
11. 15
12. 20
13. 21
14. 28
15. 30
16. 35
17. 42
18. 60
19. 70
20. 84
21. 105
22. 140
23. 210
24. 420

Now, we check which of these factors plus 1 is divisible by 3:
- \( 1 + 1 = 2 \) (not divisible by 3)
- \( 2 + 1 = 3 \) (divisible by 3)
- \( 3 + 1 = 4 \) (not divisible by 3)
- \( 4 + 1 = 5 \) (not divisible by 3)
- \( 5 + 1 = 6 \) (divisible by 3)
- \( 6 + 1 = 7 \) (not divisible by 3)
- \( 7 + 1 = 8 \) (not divisible by 3)
- \( 10 + 1 = 11 \) (not divisible by 3)
- \( 12 + 1 = 13 \) (not divisible by 3)
- \( 14 + 1 = 15 \) (divisible by 3)
- \( 15 + 1 = 16 \) (not divisible by 3)
- \( 20 + 1 = 21 \) (divisible by 3)
- \( 21 + 1 = 22 \) (not divisible by 3)
- \( 28 + 1 = 29 \) (not divisible by 3)
- \( 30 + 1 = 31 \) (not divisible by 3)
- \( 35 + 1 = 36 \) (divisible by 3)
- \( 42 + 1 = 43 \) (not divisible by 3)
- \( 60 + 1 = 61 \) (not divisible by 3)
- \( 70 + 1 = 71 \) (not divisible by 3)
- \( 84 + 1 = 85 \) (not divisible by 3)
- \( 105 + 1 = 106 \) (not divisible by 3)
- \( 140 + 1 = 141 \) (divisible by 3)
- \( 210 + 1 = 211 \) (not divisible by 3)
- \( 420 + 1 = 421 \) (not divisible by 3)

### Step 6: Count the valid factors
The valid factors of 420 that can be expressed as \( 3n - 1 \) are:
- 2
- 5
- 14
- 20
- 35
- 140

Thus, there are **6** such factors.

### Conclusion
The total number of terms in the sequence that are integers is **6**.