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In mathematics, a **recursive formula** is a way to define the terms of a sequence with respect to previous terms. It's like an instruction manual that tells you how to calculate the next step based on the one before it. This is especially helpful in finding patterns and can make calculations much easier once you understand the base rules.

For example, in our exercise, the sequence is defined using a recursive formula. The first term is given as \( a_1 = 200 \). The formula for the subsequent terms is defined as \( a_{n+1} = a_n - 10 \). This tells us that to find any term in the sequence, you subtract 10 from the term before it.

By understanding recursive formulas, you can easily generate further terms of a sequence without recalculating everything from scratch. You start with the first term and apply the same rule repeatedly.

**Sequence terms** refer to the items within a sequence. In an arithmetic sequence like the one from the exercise, each term is related and can be generated using a formula. This particular sequence is defined starting from \( a_1 = 200 \).

To find every successive term, apply the recursive rule: subtract 10 from the previous term's value:

• **First Term**: \( a_1 = 200 \)
• **Second Term**: \( a_2 = a_1 - 10 = 200 - 10 = 190 \)
• **Third Term**: \( a_3 = a_2 - 10 = 190 - 10 = 180 \)
• **Fourth Term**: \( a_4 = a_3 - 10 = 180 - 10 = 170 \)
• **Fifth Term**: \( a_5 = a_4 - 10 = 170 - 10 = 160 \)

Sequence terms are crucial because they form the building blocks of the sequence. Knowing how to calculate them helps in understanding the structure and behavior of the sequence.

The **common difference** is a fundamental concept in arithmetic sequences. It is the consistent difference between successive terms in the sequence. This value remains constant, which is why it is called "common."

In our current exercise, the common difference is \(-10\). You determine it by observing that each term is obtained by subtracting 10 from the previous term, hence:

• From \( a_1 \) to \( a_2 \), the difference is \( 190 - 200 = -10 \).
• From \( a_2 \) to \( a_3 \), the difference is \( 180 - 190 = -10 \).
• And so on for the other terms.

Recognizing a common difference is useful as it allows predicting future terms of a sequence easily. When you know the common difference, you can quickly calculate any term with just the first term and the sequence rule.