Question

The following sequence is arithmetic. Write two equations that represent the common difference and then find the common difference.

A) 3 + x, 5 + 2x, 2 + 4x; Common difference: 2x - 3

B) 3 + x, 5 + 2x, 2 + 4x; Common difference: x - 3

C) 3 + x, 5 + 2x, 2 + 4x; Common difference: 2

D) 3 + x, 5 + 2x, 2 + 4x; Common difference: 4x

Answer 1

The common difference in the sequence (3 + x, 5 + 2x, 2 + 4x) is found by setting the difference between consecutive terms equal. It simplifies to x - 3, thus the correct answer is option B.

Step-by-step explanation:

Identifying the Common Difference in an Arithmetic Sequence

To find the common difference in the arithmetic sequence given (3 + x, 5 + 2x, 2 + 4x), we need to create two equations that represent the differences between consecutive terms:

1. The difference between the second and first term: (5 + 2x) - (3 + x) = 2 + x.
2. The difference between the third and second term: (2 + 4x) - (5 + 2x) = -3 + 2x.

Since both differences should be equal for an arithmetic sequence, setting them equal gives us the common difference:

2 + x = -3 + 2x.

Subtracting x from both sides gives us:

2 = -3 + x, which simplifies to the common difference x - 3.

Hence, the correct answer is option B) 3 + x, 5 + 2x, 2 + 4x; Common difference: x - 3.