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In an Arithmetic Progression (AP), each term increases by a constant amount, called the common difference. The formula to find the nth term of an AP (denoted as \( a_n \)) is essential for determining specific terms in a sequence. It is given by:\[ a_n = a_1 + (n-1) \cdot d \]where:

• \( a_n \) is the nth term we want to find.
• \( a_1 \) is the first term of the sequence.
• \( n \) is the term number.
• \( d \) is the common difference between consecutive terms.

For example, let's say you have an AP with a first term \( a_1 = 7 \) and a common difference \( d = 4 \). To find the nth term, simply plug these values into the formula:\[ a_n = 7 + (n-1) \cdot 4 \]Simplifying this, we get \( a_n = 4n + 3 \). Remember, this formula lets you find the value of any term in the sequence just by knowing its position.

The sum of the first \( n \) terms in an AP is one of the key formulas in arithmetic sequences. This sum provides us with the total of adding up the first \( n \) numbers in the series. The formula is represented as:\[ S_n = \frac{n}{2} (2a_1 + (n-1) \cdot d) \]Here, \( S_n \) is the sum of the first \( n \) terms, and the other variables \( a_1 \), \( n \), and \( d \) are as previously defined:

• \( a_1 \) is the first term of the sequence.
• \( n \) is the total number of terms to be added.
• \( d \) is the common difference.

For the given problem where the sum of the first \( n \) terms is \( 2n^2 + 5n \), you can equate it to the standard sum formula to find unknowns like the first term or the common difference. This process involves equating, simplifying, and comparing coefficients to solve for \( a_1 \) and \( d \). Doing this correctly should match the expression given and can reveal more about the sequence's nature.

The common difference in an Arithmetic Progression (AP) is a crucial component that dictates how the sequence's terms increase. It is defined as the difference between any two consecutive terms in the sequence. Mathematically, if you have a sequence with terms \( a_1, a_2, a_3, \ldots \), the common difference \( d \) can be calculated by:\[ d = a_2 - a_1 \]This difference remains constant throughout the sequence. In the context of the problem solved, after equating and comparing the sum of \( n \) terms to the sum formula, the common difference \( d \) was determined to be 4:

• By matching the coefficient of \( n^2 \) in both expressions, it was established that \( d = 4 \).

Knowing the common difference, along with the first term, enables you to find any term in the sequence using the nth term formula and to analyze how fast or slow the sequence grows. The consistent rate of change provided by \( d \) makes arithmetic sequences predictable and easy to work with.