Answer :
Final answer:
For the sequence defined by (aₙ) = (67 - 2n), the sum of the first 75 terms, S₇₅, is calculated to be -525 by using the sum formula for an arithmetic sequence. The provided options do not match this result, indicating a possible error in the options.
Explanation:
To calculate S₇₅ for the sequence defined by (aₙ) = (67 - 2n), we need to find the sum of the first 75 terms of the sequence.
This sequence is an arithmetic sequence, where the common difference (d) is -2, the first term (a₁) is 67, and we want to find the sum of the first 75 terms, so n=75.
The formula for the sum of the first n terms of an arithmetic sequence is:
Sₙ = n/2 * (2a₁ + (n - 1)d)
Plugging our values into the formula, we get:
S₇₅ = 75/2 * (2*67 + (75 - 1)*(-2))
S₇₅ = 37.5 * (134 - 148)
S₇₅ = 37.5 * (-14)
S₇₅ = -525
However, none of the provided options (a. 83, b. -83, c. -675, d. -1,350) match -525. It's possible there might have been a mistake in either the question options or the calculation setup. Based on the given sequence and the sum formula, the correct answer is -525.
