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Class 10MathematicsArithmetic Progressions (AP)Finding the sum of the first $n$ terms of an APExpertLevel 68
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The sum of the first \(n\) terms of an arithmetic progression is zero, where \(n\) is a positive integer. Which of the following statements is always true?

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Answer and explanation

Correct answer: \(a_n=-a_1\)

Using \(S_n=\frac{n}{2}(a_1+a_n)\), and since \(n>0\) and \(S_n=0\), we get \(a_1+a_n=0\). Hence, \(a_n=-a_1\). Neither \(d=0\) nor an even \(n\) is necessary. Exam tip: for a zero sum, check whether the first and last terms are opposites.

Related tags

Arithmetic ProgressionSum Of N TermsAp PropertiesSequence ConceptsClass 10 Mathematics

Frequently asked questions

What is the correct answer to this question?

\(a_n=-a_1\)

Why is this the correct answer?

Using \(S_n=\frac{n}{2}(a_1+a_n)\), and since \(n>0\) and \(S_n=0\), we get \(a_1+a_n=0\). Hence, \(a_n=-a_1\). Neither \(d=0\) nor an even \(n\) is necessary. Exam tip: for a zero sum, check whether the first and last terms are opposites.

Which subject and chapter does this question cover?

This is a Class 10 Mathematics question. Chapter: Arithmetic Progressions (AP). Topic: Finding the sum of the first $n$ terms of an AP.

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10 more MCQs
  1. 1If an arithmetic progression has first term \(a\), common difference \(d\), and \(n\) terms, which formula correctly represents the sum of its first \(n\) terms?
  2. 2Find the sum of the first (15) terms of the AP (4,8,12,\ldots).
  3. 3Which formula correctly represents the sum of the first \(n\) terms of an AP with first term \(a\) and common difference \(d\)?
  4. 4What is the sum of the first (25) terms of the AP (1,3,5,\ldots)?
  5. 5What is the sum of the first (12) terms of the AP (10,20,30,\ldots)?
  6. 6What will be the sum of the first (10) terms of the AP (7,14,21,\ldots)?
  7. 7If the first term of an AP is (5), the last term is (45), and the number of terms is (9), what is the sum?
  8. 8Find the sum of the first (30) terms of the AP (3,6,9,\ldots).
  9. 9What is the sum of the first (8) terms of the AP (20,18,16,\ldots)?
  10. 10If the AP is (9,13,17,\ldots), what is the sum of the first (11) terms?
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