CalculatorUser

By the way, is there any fast way to do this?

answer is 306+8/15

The formula for the sum of an infinite geometric sequence is a/1-r

1. Take that formula and make an equation from the problem.

"The sum of a certain infinite geometric series is 2"

So 2=a/1-r               (where r represents common ratio and a represents the first term

Then make another equation for the second part of the problem

"The sum of the squares of all the terms is 3"

So 3=(a^2)/1-(r^2)

this is true because when all the terms are squared, the common ratio is also squared.

2. Solve for the equations

Since we are trying to find the common ratio, solve for r.

We get 1/7 as the answer

Im not sure if this correct, will someone please check this, I hope this is not a bogus solution!

Thanks!

Use Pythagorean Theorem to find DA. So 13^2-5^2=144, and the square root of 144 is 12. So DA must be 12.

Notice that AB has the length of 15 and DA has the length of twelve. Since some right triangles have a ratio of 3:4:5, side DB must have the length of 9

Since DC+BC is 9, and DC is 5, BC is 4.

So now we know the three sides of ABC, which is 13, 15, and 4. We can use Heron's formula to figure out the area.

Heron's formula is:

A=sqrt((S)(S-A)(S-B)(S-C))

S=(a+b+c)/2

We plug it sides into the 2nd equation, S=(13+15+4)/2, and we get S=16.

Then plug your answers into the first equation. A=sqrt((16)(16-15)(16-13)(16-4)), and we get A=24.

So the area of ABC is 24

The 15th term ends with 2

This is simple pattern recognizing

the first few terms are 1, 2, 2, 4, 8, 32, 256, 8192, 2097152...

Notice how the last digits are forming a pattern, (2, 2, 4, 8, 2, 6, 2, 2, 4, 8, 2, 6, 2...

The pattern is 2 -> 2 ->  4 -> 8 ->  2 -> 6

So if the pattern repeats every 6 terms. And assuming you are trying to find the 14th term. You divide 14 by 6 and find the remainder, which is 2.

So look at the 2nd term in the pattern. which is 2.

But remember the first term is 1, so you have to count back a term

So the answer is 2

Thank You!!!

It took me much longer than it needed to solve this problem, thank you for this solution