Find the Sum of the First @NUM Terms 135 , 45 , 15

Problem

(∑_n=1^3)((a_n))* where *(a_1)=135,(a_2)=45,(a_3)=15

Solution

1. Identify the type of sequence by finding the ratio between consecutive terms.

r=45/135=1/3

r=15/45=1/3

2. Recognize that since there is a common ratio r=1/3 this is a geometric sequence with first term (a_1)=135

3. Apply the formula for the sum of the first n terms of a geometric series, (S_n)=((a_1)*(1−rn))/(1−r) where n=3

(S_3)=(135*(1−(1/3)3))/(1−1/3)

4. Simplify the expression inside the parentheses and the denominator.

(S_3)=(135*(1−1/27))/2/3

(S_3)=(135*(26/27))/2/3

5. Calculate the final value by multiplying and dividing.

(S_3)=130⋅3/2

(S_3)=135+45+15=195

Final Answer

135+45+15=195

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