Arithmetic Progression

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What is the formula for the nth term of an arithmetic progression?
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The formula is a_n = a_1 + (n - 1)d, where a_n is the nth term, a_1 is the first term, n is the term number, and d is the common difference.
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How do you find the common difference in an arithmetic sequence?
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Subtract the first term from the second term: d = a_2 - a_1.
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How do you calculate the sum of the first n terms in an arithmetic sequence?
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The sum is given by S_n = n/2 * (2a_1 + (n - 1)d).
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What is the sum of the first 10 terms if the first term is 5 and the common difference is 3?
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The sum is 185.
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If the common difference in a sequence is negative, what does this indicate?
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It indicates that the sequence is decreasing.
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Can the terms of an arithmetic progression be identical?
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Yes, if the common difference is zero.
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What do you call an arithmetic sequence with only two numbers?
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A 2-term arithmetic sequence.
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How is an arithmetic progression different from a geometric progression?
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Arithmetic progression adds a constant (common difference), while geometric progression multiplies by a constant (common ratio).
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What happens to an arithmetic sequence if all terms are multiplied by a constant?
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The sequence remains arithmetic with a scaled common difference.
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How is the midpoint of an arithmetic sequence found?
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The midpoint is the average of the first and last terms.
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If the 5th term of an arithmetic progression is 15 and the 7th term is 21, what is the common difference?
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The common difference is 3.
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What is the general form of an arithmetic sequence?
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It's expressed as: a_1, a_1+d, a_1+2d, ..., a_1+(n-1)d.
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How can you verify if a given sequence is arithmetic?
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Check if the difference between consecutive terms is constant.
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Can an arithmetic progression be infinite?
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Yes, unless restricted, it can continue indefinitely in either direction.
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What happens to the mean of an arithmetic sequence if the same number is added to each term?
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The mean increases by that number.
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