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If the first term of an AP is 67 and the common difference is -13 , find the sum of the first 20 terms.

If the first term of an AP is 6767 and the common difference is −13-13 , find the sum of the first 2020 terms.

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Full solution

Q. If the first term of an AP is 6767 and the common difference is −13-13 , find the sum of the first 2020 terms.
  1. Identify Formula: Identify the formula to find the sum of the first nn terms of an arithmetic progression (AP). The formula is Sn=n2×(2a1+(n−1)d)S_{n} = \frac{n}{2} \times (2a_{1} + (n - 1)d), where SnS_{n} is the sum of the first nn terms, a1a_{1} is the first term, nn is the number of terms, and dd is the common difference.
  2. Substitute Values: Substitute the given values into the formula. Here, a1=67a_{1} = 67, n=20n = 20, and d=−13d = -13. So, S20=202×(2×67+(20−1)×(−13))S_{20} = \frac{20}{2} \times (2 \times 67 + (20 - 1) \times (-13)).
  3. Perform Calculations: Perform the calculations inside the parentheses first. Calculate the expression 2×672 \times 67, which is 134134, and then (20−1)×(−13)(20 - 1) \times (-13), which is 19×(−13)=−24719 \times (-13) = -247.
  4. Substitute Values: Now, substitute these values back into the formula. S20=10×(134−247)S_{20} = 10 \times (134 - 247).
  5. Calculate Expression: Calculate the expression inside the parentheses. 134−247134 - 247 equals −113-113.
  6. Multiply Result: Multiply the result by 1010. S20=10×(−113)=−1130S_{20} = 10 \times (-113) = -1130.

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