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An arithmetic sequence is defined as follows: {[a_(1)=-138],[a_(i)=a_(i-1)+6]:} Find the sum of the first 35 terms in the sequence.

An arithmetic sequence is defined as follows:\newline{a1=−138ai=ai−1+6 \left\{\begin{array}{l} a_{1}=-138 \\ a_{i}=a_{i-1}+6 \end{array}\right. \newlineFind the sum of the first 3535 terms in the sequence.

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Q. An arithmetic sequence is defined as follows:\newline{a1=−138ai=ai−1+6 \left\{\begin{array}{l} a_{1}=-138 \\ a_{i}=a_{i-1}+6 \end{array}\right. \newlineFind the sum of the first 3535 terms in the sequence.
  1. Identify Terms: Identify the first term a1a_1 and the common difference dd of the arithmetic sequence. The first term a1a_1 is given as −138-138, and the common difference dd is given as 66.
  2. Find 3535th Term: Use the formula for the nth term of an arithmetic sequence, which is an=a1+(n−1)da_n = a_1 + (n - 1)d, to find the 3535th term (a35a_{35}). We have a1=−138a_1 = -138, d=6d = 6, and n=35n = 35. So, a35=−138+(35−1)×6a_{35} = -138 + (35 - 1) \times 6.
  3. Calculate 3535th Term: Calculate the 35th35^{\text{th}} term using the values from the previous step. a35=−138+(34×6)=−138+204=66a_{35} = -138 + (34 \times 6) = -138 + 204 = 66.
  4. Find Sum of 3535 Terms: Use the formula for the sum of the first nn terms of an arithmetic sequence, which is Sn=n2×(a1+an)S_n = \frac{n}{2} \times (a_1 + a_n), to find the sum of the first 3535 terms. We have n=35n = 35, a1=−138a_1 = -138, and a35=66a_{35} = 66. So, S35=352×(−138+66)S_{35} = \frac{35}{2} \times (-138 + 66).
  5. Calculate Sum: Calculate the sum using the values from the previous step. S35=352×(−72)=17.5×(−72)=−1260S_{35} = \frac{35}{2} \times (-72) = 17.5 \times (-72) = -1260.

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