Final answer:
To find the first term a₁ of the geometric series given that the ratio of successive terms is 1/2 and the sum of the series is 64, we use the sum formula for a geometric series a₁ = 64 * r, which results in a₁ being 32.
Step-by-step explanation:
The given series is such that the ratio of the (n+1)th term to the nth term is constant at 1/2. Knowing that the sum of the series ∑aₙ is 64, we can identify this as a geometric series where each term is half the previous one. To find the first term a₁, we can use the formula for the sum of an infinite geometric series, which is a₁ / (1 - r), where r is the common ratio. Since r = 1/2, we have 64 = a₁ / (1 - 1/2), which simplifies to 64 = a₁ / (1/2). Multiplying both sides by 1/2 gives us a₁ = 64 * 1/2, therefore a₁ = 32.