Use mathematical induction to prove that if a ∈ ℝ and r≠1then a+ar^1+ar^2++ar^(n+1)=a(r^n+1)/(r-1 ).

Step-by-step explanation:

In this problem we are proving the equality

.

When we want prove an equality by induction we need to follow some steps.

First: Check the hypothesis for some initial cases. In this exercise we take

: we have . So the equality holds for .

: we have . So the equality holds for .

After we have checked the hypothesis for can continue to the next step.

Second: State the induction hypothesis for . In this case the hypothesis is:

.

Now, it comes the last step and, usually, the most difficult.

Third: Prove the statement for (using that the equality holds for !). This means that we want to prove that:

.

So, let us start by the left hand side and try to get the left hand side.

.

We can group the above sum in the following way

.

Notice that the expression under parenthesis is the same we have in our induction hypothesis. Then,

.

Now, we operate the sum that appears in the right hand side:

.

So, we have obtained that

,

which is exactly what we want to prove.

find the magnitude of the trig term  

csc

(

x

)

−

1

by taking the absolute value of the coefficient.

1

find the lower bound of the range.

tap for more

y

=

−

2

find the upper bound of the range.

tap for more

y

=

0

the range is  

y

≤

−

2

or  

y

≥

0

.

(

−

∞

,

−

2

]

∪

[

0

,

∞

)

{

y

|

y

≤

−

2

,

y

≥

0

}

step-by-step explanation:

i do not know the answer

step-by-step explanation: