Use mathematical induction to prove the following statement: The sum of the first n even positive integers is (n2 + n). That is, 2 4 6 8 2n

Step-by-step explanation:

To prove that the sum of the first n even +ve integers is:

By using mathematical induction;

For n = 1, we get:

2n = 2 × 1 = 2

2 = 1² + 1 ----- (1)

∴ the outcome is true if n = 1

However, let assume that the result is also true for n = k

Now,

we can now say:

∴

Thus, the result is true for n = m+1, hence we can posit that the result is also true for each value of n.

As such

use the power rule  

( a b ) n = a n b n

to distribute the exponent.

apply the product rule to  

3 x 2 y 3 . ( 3 x 2 ) 2 ( y 3 ) 2

apply the product rule to  

3 x 2 . 3 2 ( x 2 ) 2 ( y 3 ) 2

raise  

3

to the power of  

2 . 9 ( x 2 ) 2 ( y 3 ) 2 multiply the exponents in   ( x 2 ) 2

.

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 2 ⋅ 2 ( y 3 ) 2

multiply  

2   by   2 . 9 x 4 ( y 3 ) 2

multiply the exponents in  

( y 3 ) 2 .

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 4 y 3

⋅

2

multiply  

3   by   2 . 9 x 4 y 6

step-by-step explanation:

the final price would be 29.6925

step-by-step explanation:

39.59 minus 25% equals 29.6925