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Class 10MathematicsArithmetic Progressions (AP)Finding the sum of the first $n$ terms of an APHard
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In an AP, a = 7, and the sum from the 21st term to the 40th term is 3680. Find d.

Answer and explanation

Correct answer: 6

Answer: A, d = 6. The terms from the 21st through the 40th positions contain 40 − 21 + 1 = 20 terms. The 21st term is a + 20d = 7 + 20d, and the 40th term is a + 39d = 7 + 39d. The sum of these 20 AP terms is number of terms divided by 2 multiplied by the sum of the first and last terms: 3680 = 20/2[(7 + 20d) + (7 + 39d)] = 10(14 + 59d). Divide by 10: 368 = 14 + 59d. Hence 59d = 354 and d = 6. Therefore option A is correct. Option B, C, and D fail when substituted into the same equation. The common confusion is using 20d for the last term; 20 is the number of selected terms, whereas the 40th term has 39 differences after the first term. A useful check is that S_40 − S_20 gives the same block sum.

Related tags

Arithmetic ProgressionCommon DifferenceRange SumAp FormulaFinding The Sum Of The First $N$ Terms Of An ApFinding The Sum Of The First N Terms Of An ApArithmetic Progressions (Ap)Arithmetic Progressions 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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