Question

Create a formula for the sequence 2, 3.5, 6.125, 10.71875

Answer 1

To find a formula for the sequence \(2, 3.5, 6.125, 10.71875\), we first examine the terms in detail:

1. **First Term (\(n=1\))**: \(a_1 = 2\)
2. **Second Term (\(n=2\))**: \(a_2 = 3.5\)
3. **Third Term (\(n=3\))**: \(a_3 = 6.125\)
4. **Fourth Term (\(n=4\))**: \(a_4 = 10.71875\)

Let's look at the differences between the terms:

- \(a_2 - a_1 = 3.5 - 2 = 1.5\)
- \(a_3 - a_2 = 6.125 - 3.5 = 2.625\)
- \(a_4 - a_3 = 10.71875 - 6.125 = 4.59375\)

Next, let's calculate the ratios of consecutive terms to see if there is a multiplicative pattern:

- \(\frac{a_2}{a_1} = \frac{3.5}{2} = 1.75\)
- \(\frac{a_3}{a_2} = \frac{6.125}{3.5} = 1.75\)
- \(\frac{a_4}{a_3} = \frac{10.71875}{6.125} \approx 1.75\)

It seems that each term is obtained from the previous term by multiplying by \(1.75\).

Based on this, we can express the \(n\)-th term \(a_n\) as:
\[
a_n = a_1 \cdot (1.75)^{n-1}
\]

Thus,
\[
a_n = 2 \cdot (1.75)^{n-1}
\]

Let's verify this formula with the terms of the sequence:

1. For \(n=1\):
\[
a_1 = 2 \cdot (1.75)^{0} = 2 \cdot 1 = 2
\]
2. For \(n=2\):
\[
a_2 = 2 \cdot (1.75)^{1} = 2 \cdot 1.75 = 3.5
\]
3. For \(n=3\):
\[
a_3 = 2 \cdot (1.75)^{2} = 2 \cdot 3.0625 = 6.125
\]
4. For \(n=4\):
\[
a_4 = 2 \cdot (1.75)^{3} = 2 \cdot 5.37890625 \approx 10.71875
\]

This confirms that our formula is correct.

Hence, the formula for the sequence is:
\[
a_n = 2 \cdot (1.75)^{n-1}
\]

Answer 2

Lori buys 5 goldfish for her tank her research shows that they will reproduce at a rate of 32% each month which statement discusses a key feature of the graph of this situation