Determine the no. of terms in the geometric sequence {81, 27, 9 …...1/81} Also find their associated sum.

To determine the number of terms in the given geometric sequence, we need to find the common ratio first. The common ratio can be found by dividing any term by its previous term.

Let's take the second term, 27, and divide it by the first term, 81:

27 / 81 = 1/3

So, the common ratio is 1/3.

Now, we can find the number of terms using the formula for the nth term of a geometric sequence:

nth term = a * r^(n-1)

In this case, the nth term is 1/81, the first term is 81, and the common ratio is 1/3. Plugging these values into the formula, we get:

1/81 = 81 * (1/3)^(n-1)

To simplify the equation, we can rewrite 1/81 as (1/3)^4:

(1/3)^4 = 81 * (1/3)^(n-1)

Now, we can equate the exponents:

4 = n - 1

Solving for n, we find:

n = 5

Therefore, the geometric sequence has 5 terms.

To find the sum of the terms, we can use the formula for the sum of a geometric series:

sum = a * (1 - r^n) / (1 - r)

In this case, the first term is 81, the common ratio is 1/3, and the number of terms is 5. Plugging these values into the formula, we get:

sum = 81 * (1 - (1/3)^5) / (1 - 1/3)

Simplifying the equation, we find:

sum = 81 * (1 - 1/243) / (2/3)

sum = 81 * (242/243) / (2/3)

sum = 81 * (242/243) * (3/2)

sum = 81 * 242 * 3 / (243 * 2)

sum = 729 / 2

Therefore, the sum of the terms in the geometric sequence is 729/2.