Use the method of successive differences to determine the next number in the given sequence.

Assuming that the pattern continues, the eighth term would be . (Simplify your answer.)

Ask by Jimenez Reeves. in Mexico

Jan 23,2025

Question

Use the method of successive differences to determine the next number in the given sequence.

Assuming that the pattern continues, the eighth term would be . (Simplify your answer.)

Ask by Jimenez Reeves. in Mexico

Jan 23,2025

UpStudy AI · Accepted Answer

To determine the next number in the sequence $$ -2, -4, -5, -5, -4, -2, 1 $$ using the **method of successive differences**, follow these steps:

1. **Compute the First Differences:**
   Subtract each term from the subsequent term.
   
   $$
   \begin{align*}
   -4 - (-2) &= -2 \
   -5 - (-4) &= -1 \
   -5 - (-5) &= 0 \
   -4 - (-5) &= 1 \
   -2 - (-4) &= 2 \
   1 - (-2) &= 3 \
   \end{align*}
   $$
   
   **First Differences:** $$ -2, -1, 0, 1, 2, 3 $$

2. **Compute the Second Differences:**
   Subtract each first difference from the subsequent first difference.

$$
   \begin{align*}
   -1 - (-2) &= 1 \
   0 - (-1) &= 1 \
   1 - 0 &= 1 \
   2 - 1 &= 1 \
   3 - 2 &= 1 \
   \end{align*}
   $$
   
   **Second Differences:** $$ 1, 1, 1, 1, 1 $$

Since the second differences are **constant**, this indicates that the original sequence can be modeled by a quadratic (degree 2) polynomial.

3. **Determine the Next First Difference:**
   Since the second differences are constant (all are 1), the next first difference will be:

$$
   3 + 1 = 4
   $$

4. **Find the Next Term in the Sequence:**
   Add the next first difference to the last term in the original sequence.

$$
   1 + 4 = 5
   $$

**Therefore, the eighth term in the sequence is $$ \boxed{5} $$.**
The eighth term in the sequence is 5.

Use the method of successive differences to determine the next number in the given sequence.

Assuming that the pattern continues, the eighth term would be . (Simplify your answer.)

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Use the method of successive differences to determine the next number in the given sequence. Assuming that the pattern continues, the eighth term would be . (Simplify your answer.)

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Use the method of successive differences to determine the next number in the given sequence.

Assuming that the pattern continues, the eighth term would be . (Simplify your answer.)