Question

for an arithmetic sequence, (a_{9} = 6). if the common difference is -1, find:
(a_{1} =)
the sum of the first 68 terms =

Explanation:

Step1: Identify given values

$$a_{9} = 6, \quad d = -1$$

Step2: Use arithmetic sequence formula

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

Step3: Substitute values to find \(a_{1}\)

$$6 = a_{1} + (9 - 1)(-1)$$

Step4: Solve for \(a_{1}\)

$$a_{1} = 14$$

Step5: Use arithmetic sum formula

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

Step6: Substitute values for \(S_{68}\)

$$S_{68} = \frac{68}{2} [2(14) + (68 - 1)(-1)]$$

Step7: Calculate the final sum

$$S_{68} = 34 [28 - 67] = -1326$$

Answer:

\(a_{1} = 14\)
the sum of the first 68 terms = \(-1326\)