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An arithmetic sequence is defined as follows: {[a_(1)=92],[a_(i)=a_(i-1)-8]:} Find the sum of the first 28 terms in the sequence.

An arithmetic sequence is defined as follows:\newline{a1=92ai=ai−1−8 \left\{\begin{array}{l} a_{1}=92 \\ a_{i}=a_{i-1}-8 \end{array}\right. \newlineFind the sum of the first 2828 terms in the sequence.

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Q. An arithmetic sequence is defined as follows:\newline{a1=92ai=ai−1−8 \left\{\begin{array}{l} a_{1}=92 \\ a_{i}=a_{i-1}-8 \end{array}\right. \newlineFind the sum of the first 2828 terms in the sequence.
  1. Identify terms and difference: Identify the first term a1a_1 and the common difference dd of the arithmetic sequence.\newlineThe first term a1a_1 is given as 9292, and the common difference dd is the amount subtracted from each term to get the next, which is 88.
  2. Use sum formula: Use the formula for the sum of the first nn terms of an arithmetic sequence, which is Sn=n2∗(2a1+(n−1)d)S_n = \frac{n}{2} * (2a_1 + (n - 1)d), where SnS_n is the sum of the first nn terms, a1a_1 is the first term, dd is the common difference, and nn is the number of terms.
  3. Plug values and calculate: Plug the values into the formula to find the sum of the first 2828 terms.\newlineWe have n=28n = 28, a1=92a_1 = 92, and d=−8d = -8 (since the sequence is decreasing).\newlineS28=282×(2×92+(28−1)×(−8))S_{28} = \frac{28}{2} \times (2\times92 + (28 - 1)\times(-8))
  4. Simplify expression: Simplify the expression inside the parentheses first.\newline2×92=1842\times 92 = 184\newline(28−1)×(−8)=27×(−8)=−216(28 - 1)\times(-8) = 27\times(-8) = -216\newlineNow add these two results: 184+(−216)=−32184 + (-216) = -32
  5. Calculate sum: Now, calculate the sum using the simplified expression.\newlineS28=282×(−32)S_{28} = \frac{28}{2} \times (-32)\newlineS28=14×(−32)S_{28} = 14 \times (-32)\newlineS28=−448S_{28} = -448

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