Consider the sequence 1, – 2, 3, – 4, 5, – 6, . . . . . . n{( – 1)^{n + 1}}. What is the average of the first 300 terms of the sequence? (a) – 1 (b) 0.5 (c) 0 (d) – 0.5

To solve this problem, let's first understand the structure of the sequence and then find the average of the first 300 terms.

The given sequence is:

1, -2, 3, -4, 5, -6, ...

This sequence alternates between positive and negative integers. The general term of the sequence is given by:

a_n = n * (-1)^(n+1)

This expression defines the alternating sign pattern. For odd values of n (n = 1, 3, 5, ...), the term will be positive, and for even values of n (n = 2, 4, 6, ...), the term will be negative.

Step 1: Determine the sum of the first 300 terms
We need to calculate the sum of the first 300 terms of the sequence. Notice that the sequence alternates, so we can group terms in pairs:

(1, -2), (3, -4), (5, -6), ...

Each pair of terms has the form (odd number, negative even number). The sum of each pair is:

1 + (-2) = -1
3 + (-4) = -1
5 + (-6) = -1
... and so on.

Thus, each pair of consecutive terms adds up to -1. Since we are looking at the first 300 terms, we can divide these terms into 150 pairs (because 300 terms / 2 terms per pair = 150 pairs). Each pair contributes -1 to the sum.

The total sum of the first 300 terms is:

Sum = 150 * (-1) = -150

Step 2: Find the average of the first 300 terms
The average of the first 300 terms is given by the formula:

Average = (Sum of terms) / (Number of terms)

Substituting the sum and the number of terms:

Average = (-150) / 300 = -0.5

Final Answer:
The average of the first 300 terms is -0.5.

Thus, the correct option is (d) -0.5.