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Finding the number of terms in an arithmetic sequence might sound like a complex task, but it’s actually pretty straightforward. All you need to do is plug the given values into the formula t_{n} = a + (n  1) d and solve for n, which is the number of terms. Note that t_{n} is the last number in the sequence, a is the first term in the sequence, and d is the common difference.
Steps

1Identify the first, second, and last terms of the sequence. Typically, to solve a problem like this, you’ll be given the first 3 or more terms as well as the last term.^{[1] X Research source }
 For example, you may have the following sequence: 107, 101, 95…61. In this case, the first term is 107, the second term is 101, and the last term is 61. You need all of this information to solve the problem.

2Subtract the first term from the second term to find the common difference. In the example sequence, the first term is 107 and the second term is 101. So, subtract 107 from 101, which is 6. Therefore, the common difference is 6.^{[2] X Research source }Advertisement

3Use the formula t_{n} = a + (n  1) d to solve for n. Plug in the last term (t_{n}), the first term (a), and the common difference (d). Work through the equation until you’ve solved for n.^{[3] X Research source }
 For example, start by writing: 61 = 107 + (n  1) 6. Subtract 107 from both sides so you’re left with 168 = (n  1) 6. Then, divide both sides by 6 to get 28 = n  1. Finish by adding 1 to both sides so that n = 29.
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Community Q&A

QuestionCan I use the formula (A  L/d) + 1 for finding n?DonaganTop AnswererAs explained above, n = [(L  A) / d] + 1.

QuestionIf the difference is not given, how do I find the difference?DonaganTop AnswererSubtract any term from the one that follows it.

QuestionIf the first term and last term of an arithmetic progression are 5 and 89, how do I find the number of terms?DonaganTop AnswererYou can't without knowing the difference between consecutive terms.

QuestionWhat is the common difference in a minus?Community AnswerHere is an example: 2, 5, 8, 11. In this sequence the common difference (d) is found by subtracting any term from the term that precedes it. For instance, 5  (2) = 5 + 2 = 3. The common difference is a negative 3, which means 3 is always subtracted from a given term to find the next term in the sequence.

QuestionDoes an arithmetic sequence start at n=0?Community AnswerWhen given a sequence, you'd usually get the first term. Unless n(1)=0, then no, the arithmetic sequence doesn't always begin with 0.

QuestionHow do I find the term number? Can someone explain in an easy, simple way?DonaganTop AnswererTo find the number of terms in an arithmetic sequence, divide the common difference into the difference between the last and first terms, and then add 1.

QuestionHow can I find number of items when arithmetic mean is given but sum of items is not given?DonaganTop AnswererYou would need more information, such as the common difference and the first and last terms.

QuestionHow many terms are there in 31, 32, 33, 47, 48, 49?Community AnswerAssuming this sequence consists of every integer from 31 through 49, subtract 31 from 49, then add 1.

QuestionI've been given this sequence: 2+10+18. How do I find the sum of the first 40 terms?DonaganTop AnswererFinding the sum of an arithmetic sequence involves finding the average of the first and last numbers of the sequence. Therefore, you must know the 40th term. Once you find the 40th term (there's a wikiHow article on finding a certain term in an arithmetic sequence), add it to 2, divide by 2, then multiply by 40. That's the sum you're looking for.

QuestionI've been given the sum of the sequence, the first term, and the common difference. How can I get the number of terms?DonaganTop AnswererYou don't have enough information to find the number of terms quickly. However, you could start at the first term and keep adding the common difference over and over until you reach the given sum of the sequence. Count the number of times you added the common difference. Add 1 to that number to get the number of terms in the sequence.
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Tips
 The difference between the last term and first term will always be divisible by the common difference.Thanks!
Warnings
 Do not confuse the difference between the first and last term with the common difference.Thanks!