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The first term of an arithmetic sequence is simply the starting number of the sequence. It is crucial because it sets the baseline for the entire sequence.
In the sequence given: \[7, 3, -1, -5, \dots \] The first term is the very first number you see, which is 7.
Identifying the first term is straightforward as it is always the initial value of the sequence.
In any arithmetic sequence, knowing the first term is important because it helps in calculating the other terms and any related mathematics that follow. Without the first term, you can't start or build the sequence.

The common difference (often represented as \(d\)) is what makes an arithmetic sequence unique. It is the amount that each term in the sequence increases or decreases by moving from one term to the next. You find the common difference by subtracting the first term from the second term:
\[ d = 3 - 7 = -4 \]
This example tells us that each term in the sequence decreases by 4. Here are the steps to determine the common difference:

• Take the second term and subtract the first term.
• Verify this by subtracting the next term from the previous term.

This ensures that the difference remains the same. Let's check another term to confirm:
\[ d = -1 - 3 = -4 \]
You see, the difference does stay the same as it is -4.
Always ensure that the difference is constant to establish it as an arithmetic sequence.

In an arithmetic sequence, the common difference should always be consistent. This means that the difference between consecutive terms doesn't change.
Having a consistent difference is crucial to maintain the property of the arithmetic sequence. To check if the difference is consistent throughout the sequence, you subtract each pair of terms, and they should all be the same:
\[ d = -1 - 3 = -4 \]
And another example from the sequence:
\[ d = -5 - (-1) = -5 + 1 = -4 \]
By ensuring the difference is consistent across terms, you affirm that the entire sequence follows the arithmetic pattern. This consistency helps in predicting any further terms in the sequence.
Consistency in the common difference is a key characteristic of arithmetic sequences, making them easier to analyze and understand.