Final answer:
To compute (2 + 4 + 6 + ... + 36)/(3 + 6 + 9 + ... + 54), each series is summed using the arithmetic series formula and the result is simplified to get 2/3.
Step-by-step explanation:
To compute the given series (2 + 4 + 6 + ... + 36) / (3 + 6 + 9 + ... + 54), we should first recognize that both the numerator and the denominator are arithmetic series. The first series is even numbers starting from 2 and ending at 36, and the second series is multiples of 3 starting from 3 and ending at 54.
To find the sum of an arithmetic series, the formula is:
S = n/2 (a1 + an)
where S is the sum of the series, n is the number of terms, a1 is the first term, and an is the last term.
For the numerator (even numbers from 2 to 36), we can see that the first term a1 is 2, and the last term an = 36. The number of terms n can be calculated as (36/2) = 18. So, the sum of the numerator series is S = 18/2 (2 + 36) = 9 * 38 = 342.
For the denominator (multiples of 3 from 3 to 54), the first term a1 is 3, and the last term an = 54. The number of terms n = 54/3 = 18. Thus, the sum of the denominator series is S = 18/2 (3 + 54) = 9 * 57 = 513.
Now, we divide the sum of the numerator series by the sum of the denominator series: 342 / 513. Simplifying this, we get 2/3. So, the computed value of the given expression is 2/3.