Question

for an arithmetic sequence, (a_{37} = 82). if the common difference is -1, find:
(a_1 =)
the sum of the first 30 terms =

Explanation:

Step1: Identify the arithmetic sequence formula

$$a_{n} = a_{1} + (n - 1)d$$

Step2: Substitute given values to find \(a_{1}\)

$$82 = a_{1} + (37 - 1)(-1)$$

Step3: Solve for the first term

$$a_{1} = 118$$

Step4: Identify the sum formula

$$S_{n} = \frac{n}{2} [2a_{1} + (n - 1)d]$$

Step5: Substitute values to find \(S_{30}\)

$$S_{30} = \frac{30}{2} [2(118) + (30 - 1)(-1)]$$

Step6: Calculate the final sum

$$S_{30} = 15 [236 - 29] = 3105$$

Answer:

\(a_{1} = 118\)
the sum of the first 30 terms = \(3105\)