Given the geometric sequence that begins with the terms 2,1,... What is the sum of the first 12 terms?

The general term of a geometric series is arn−1, where r is the ratio, and a is the first term. In your case, a=2–√ and r=3–√. The 12th term is then

2–√(3–√)11=2436–√

The sum of the first 12 terms is

a∑k=011rk=ar12−1r−1=2–√7283–√−1=364(2–√+6–√)

Note that I used 35=243 and 36=729 in the above.

EDIT

Note that

2–√7283–√−1=2–√7283–√−13–√+13–√+1=2–√(3–√+1)728(3–√)2−12=(6–√+2–√)7283−1

Step-by-step explanation:

20

step-by-step explanation:

0.16 or 16%

step-by-step explanation:

lets take a coin for example.

if you toss it twice, it has a 1/4 chance of landing on heads both times because 1/2 * 1/2 is 1/4.

with the cards, the chance of picking dog is 0.4. having two people pick one is like picking a card twice in a row. so 0.4 * 0.4 is 0.16, or 16%