Question

What kind of sequence is this?

85, –85, 170, –510, ...

Answer 1

To determine the type of sequence given (85, -85, 170, -510, ...), let's analyze the terms.

1. **Identifying the pattern:**
- The first term is \( 85 \).
- The second term is \( -85 \), which can be seen as \( 85 \times (-1) \).
- The third term is \( 170 \), which is \( 85 \times 2 \).
- The fourth term is \( -510 \), which is \( 85 \times (-6) \).

2. **Calculating the ratios:**
- The ratio from the first to the second term: \( -85/85 = -1 \)
- The ratio from the second to the third term: \( 170/(-85) = -2 \)
- The ratio from the third to the fourth term: \( -510/170 = -3 \)

The multiplicative factors are as follows:
- From the first to the second, the factor is \(-1\).
- From the second to the third, the factor is \(-2\).
- From the third to the fourth, the factor is \(-3\).

3. **Finding a general term:**
Now we can express the \(n^{th}\) term (starting from \(n=0\)):
- \(a_0 = 85\)
- \(a_1 = 85 \cdot (-1)^1\)
- \(a_2 = 85 \cdot 2(-1)^0\)
- \(a_3 = 85 \cdot 3(-1)^1\)

Checking for a recognized structure:
It's evident that the terms seem to involve alternating signs and increasing magnitudes:

According to the established factors, this sequence can be described as:
- \( a_n = 85 \cdot n(-1)^{n} \)

In conclusion, this is a sequence characterized by both alternating signs and multiplication by incrementally increasing integers.