Question

The successive differences for a sequence are shown. What is the standard form polynomial that represents the sequence?

6 __ 23 __ 46 __ 75 __ 110
17 __ 23 __ 28 __ 35
6 6 6


A. 2x^2 + 9x + 7
B. 4x^2 - x + 12
C. 3x^2 + 8x - 5
D. 2x^2 + 4x - 3

Answer 1

To find the standard form polynomial that represents the given sequence, we can use the information from the successive differences.

The original sequence is:
- 6
- 23
- 46
- 75
- 110

The first differences are:
- 23 - 6 = 17
- 46 - 23 = 23
- 75 - 46 = 29
- 110 - 75 = 35

The second differences are:
- 23 - 17 = 6
- 29 - 23 = 6
- 35 - 29 = 6

Since the second differences are constant (equal to 6), this indicates that the original sequence can be represented by a quadratic polynomial of the form \( ax^2 + bx + c \).

We start with the second difference:
- The second difference being constant and equal to \( 6 \) means \( 2a = 6 \).
- So, \( a = 3 \).

Next, we use the information from the sequence to find coefficients \( b \) and \( c \). We know three points from the sequence:
1. When \( x = 1 \), \( f(1) = 6 \)
2. When \( x = 2 \), \( f(2) = 23 \)
3. When \( x = 3 \), \( f(3) = 46 \)

Using the polynomial form \( f(x) = 3x^2 + bx + c \):

1. For \( x = 1 \):
\[
3(1)^2 + b(1) + c = 6
\]
\[
3 + b + c = 6 \quad \text{(1)}
\]
\[
b + c = 3 \quad \text{(simplified)}
\]

2. For \( x = 2 \):
\[
3(2)^2 + b(2) + c = 23
\]
\[
3(4) + 2b + c = 23
\]
\[
12 + 2b + c = 23
\]
\[
2b + c = 11 \quad \text{(2)}
\]

3. For \( x = 3 \):
\[
3(3)^2 + b(3) + c = 46
\]
\[
3(9) + 3b + c = 46
\]
\[
27 + 3b + c = 46
\]
\[
3b + c = 19 \quad \text{(3)}
\]

Now we can solve the system of equations made from equations (1), (2), and (3).

From (1):
\[
c = 3 - b
\]

Substituting \( c \) into (2):
\[
2b + (3 - b) = 11
\]
\[
2b + 3 - b = 11
\]
\[
b + 3 = 11
\]
\[
b = 8
\]

Now substituting \( b = 8 \) back into (1):
\[
8 + c = 3
\]
\[
c = 3 - 8 = -5
\]

Thus, we have found:
- \( a = 3 \)
- \( b = 8 \)
- \( c = -5 \)

The polynomial that represents the sequence is:
\[
f(x) = 3x^2 + 8x - 5
\]

Therefore, the standard form polynomial that represents the sequence is:

**C. 3x² + 8x - 5**.