Evaluate the summation of 20 times 0.5 to the n minus 1 power, from n equals 3 to 12.

The sum of the series is 9.99

Step-by-step explanation:

We are given the series .

So, , where n=3 to 12

Then, we have,

Thus, the common ratio is

Since, the sum of first n terms of a series is .

As n = 3 to 12, then the number of terms = 10, first term=a₃= 5 and r= 0.5

So, the sum of 10 terms is

i.e.

i.e.

i.e.

i.e. S = 9.99

Hence, the sum of the series is 9.99

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(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240