Report error find the sum of the series 1 + \frac{1}{2} + \frac{1}{10} + \frac{1}{20} + \frac{1}{100} + \cdots, where we alternately multiply by \frac 12 and \frac 15 to get successive terms.

The sum of given  series is 5/3.

Step-by-step explanation:

Given series is

We need to find the sum of above series. In this series we alternately multiply by  1/2 and 1/5 to get successive terms.

Isolate the odd and even place terms.

Now, we have two infinite series.

Sum of an infinite GP is

where, r is first term and r is common ratio, 0 < r < 1.

In

First term = 1

Common ratio = 1/10

In

First term = 1/2

Common ratio = 1/10

The sum of given series is

Therefore the sum of given  series is 5/3.

3. parallelogram

step-by-step explanation:

we turn each of our sides into direction vectors

ab=b-a=(3 - 7, 1 - 0) = (-4, 1)

bc=c-b=(1,6)

cd=d-c=(4,-1)

da=a-d=(-1,-6)

we see how the opposite sides have negated direction vectors; this is characteristic of a parallelogram.   we have congruent sides ab=cd and bc=da with unequal lengths because 1^2+6^2 is different from (-4)^2+1^2.   so it's not a square or a rhombus.  

we check the dot product of direction vectors of adjacent sides.   we have

ab dot bc = -4(1) + 1(6) = 2

that's not zero so the sides are not perpendicular.   we've ruled out rectangle.

the direction vectors indicate two pairs of congruent parallel sides, so we choose parallelogram over trapezoid.

first anwser choice.

step-by-step explanation:

the greater/less or equal to symbol (> or < with the line at the bottom) is a closed/filled in circle.

-5c + 2 _< _27

subtract from both sides by 2

-5c _< _ 27

divide both sides by -5

c _< _ 5.4

flip the inequality sign

c _> _5.4

the only possible answer that would match the simplified inequality is the first answer choice