To solve the problem, we first observe the pattern of the sequence (\sin\left(\frac{n\pi}{2}\right)) for (n=1,2,3,4,\ldots):

Step 1: Identify the repeating pattern

Calculate the values of (\sin\left(\frac{n\pi}{2}\right)):

• (n=1): (\sin\left(\frac{\pi}{2}\right)=1)
• (n=2): (\sin(\pi)=0)
• (n=3): (\sin\left(\frac{3\pi}{2}\right)=-1)
• (n=4): (\sin(2\pi)=0)
• (n=5): (\sin\left(\frac{5\pi}{2}\right)=1) (cycle repeats)

The pattern is ([1,0,-1,0]), repeating every 4 terms.

Step 2: Sum of one cycle

Each cycle of 4 terms sums to:
(1 + 0 + (-1) + 0 = 0)

Step 3: Number of cycles in 100 terms

(100 \div 4 = 25) full cycles.

Total sum = (25 \times 0 = 0)

Answer: (\boxed{0})

作者声明:本文包含人工智能生成内容。