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.

1. is 4 and 2. is 3

step-by-step explanation:

so if you look at the first one you see that 1 is to 4 and 2 is to 8 and so on, so you can see for every 1 there is a times 4 so if it were 3 then it would be 3*4 and it is 12

for number 2 it is 3 because 1 is to 3 and 2 is to 6 so for every 1 there is 3 so if it were 3 then there would be 3*3 = 9