Final answer:
To find the first term of a geometric series with a sum to infinity of 6 and a ratio of ½, use the formula S = a / (1 - r). Solving for 'a' gives the first term as 3.
Step-by-step explanation:
To calculate the first term of the geometric series with a constant ratio of ⅔ and a sum to infinity of 6, we use the formula for the sum to infinity of a geometric series: S = a / (1 - r), where S is the sum to infinity, a is the first term, and r is the common ratio. Given that S = 6 and r = ⅔, the formula becomes 6 = a / (1 - ⅔). Solving for a, we multiply both sides by (1 - ⅔), which gives us a = 6 × (1 - ⅔) = 6 × ½ = 3.
Therefore, the first term of the series (a) is 3