Question

For the given statement Pn, write the statements P1, Pk, and Pk+1.

2 + 4 + 6 + . . . + 2n = n(n+1)

Answer 1

what is the setup when n=1?

Answer 2

what do you mean?

Answer 3

2n = n(n+1) , when n=1

then replace n with k in the setup to define when n=k

Answer 4

the last step will be an induction step

using the rule:

2+4+6+....+2n ; when n=k+1 is

2+4+6+....+2(k+1), or better yet

2+4+6+....+2k+2(k+1)

Pk + 2(k+1) = k(k+1) + 2(k+1)

Answer 5

work the right side to match the format of: n(n+1), for n=k+1

Answer 6

??

Answer 7

im not real sure what your confusion is .....

Answer 8

i dont get any of this
how do i do for p1?

Answer 9

replace n with a 1

Answer 10

then pk2

Answer 11

pk*

Answer 12

the pk+1

Answer 13

P(n) = 2 + 4 + 6 + ... + 2n = n(n+1)

P(1) = 2(1) = 1(1+1) ; is true

Answer 14

for Pk, replace n with a k

P(n) = 2 + 4 + 6 + ... + 2n = n(n+1)

P(k) = 2 + 4 + 6 + ... + 2k = k(k+1)

Answer 15

for P(k+1), replace n with a (k+1) ; but we have to show that the right side doesnt change its format

P(n) = 2 + 4 + 6 + ... + 2n = n(n+1)

P(k+1) = 2 + 4 + 6 + ... + 2k + 2(k+1) ?=? (k+1)((k+1)+1)

Answer 16

from P(k) we know that:

P(k) = 2 + 4 + 6 + ... + 2k = k(k+1)

therefore, P(k+1) is just

P(k+1) = 2 + 4 + 6 + ... + 2k + 2(k+1) = k(k+1) + 2(k+1)
^^^^^ ^^^^^^
adding 2(k+1) to P(k)

P(k+1) = P(k) + 2(k+1) = k(k+1) + 2(k+1)

show that :

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

Answer 17

thank you <3

Answer 18

youre welcome