Given the sequence: 4, 2, 2, 2, 3, 6, 15, 45, ...

If $(2835)^n$ is a term of the sequence where $n$ is the position of the term in the sequence, find the value of $n$.

Question

Given the sequence: 4, 2, 2, 2, 3, 6, 15, 45, ...

If $(2835)^n$ is a term of the sequence where $n$ is the position of the term in the sequence, find the value of $n$.

Accepted Answer

**Solution:**

1. Analyze the given sequence:
   $$4, 2, 2, 2, 3, 6, 15, 45, ...$$

2. Observe the pattern of multiplication between terms starting from the first term:
   - $4 	o 2$: multiply by $0.5$
   - $2 	o 2$: multiply by $1$
   - $2 	o 2$: multiply by $1$
   - $2 	o 3$: multiply by $1.5$
   - $3 	o 6$: multiply by $2$
   - $6 	o 15$: multiply by $2.5$
   - $15 	o 45$: multiply by $3$

3. The pattern of multipliers is increasing by $0.5$ starting from $0.5$:
   $$0.5, 1, 1.5, 2, 2.5, 3, ...$$

4. Next multipliers will be:
   - $45 	imes 3.5 \approx 157.5$
   - $157.5 	imes 4 \approx 630$
   - $630 	imes 4.5 \approx 2835$

5. The term $2835$ appears at the 10th term of the sequence (since $45$ is the 8th term, $157.5$ is 9th, and $2835$ is 10th).

**Answer:** 10th

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