use mathematical induction to prove the statement is true for all positive integers n. 6 + 12 + 18 + ... + 6n = 3n(n + 1)​

Step-by-step explanation:

6+12+18+6n=3n(n+1)

Proof by induction

True for n = 1

6 = 3(1)(1+1)

Assume true for n = k

6+12+18+6k=3k(k+1)

Then show that

6+12+18+6(k+1)= 3(k+1)(k+1+1)

We add 6(k+1) to both sides of

6+12+18+6k=3k(k+1)

6+12+18+6k +6(k+1) =3k(k+1) + 6(k+1)

= 3(k+1)(k + 2)

= 3(k+1)(k+1+1)

So true for n = k+1

Since true for n=1 it is true for n=2,3,...

Thus true for all natural numbers n by mathematical induction

l.jnl.j/n.m,lm,

step-by-step explanation:

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: