Question

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

2 + 6 + 14 + . . . + 2n = 2(n+1) - 2

Answer 1

@helder_edwin

Answer 2

r u sure?

Answer 3

if n=1 then 2n=2 (the first term)

but if n=2 then 2n=2*2=4 not the second term !!!

Answer 4

So what would we say for that?

Answer 5

it doesn't ask u to prove it. does it??

Answer 6

Well it just says to write the statement.

Answer 7

OK. then let's ignore that issue then.

but don't forget to do what i did whenever u r asked to prove this kind of stuff. OK?

Answer 8

Okay I won't

Answer 9

u have
\[ \large P(n):\quad 2+6+14+\dots+2n=2(n+1)-2 \]

Answer 10

do u know what P(1) means ??

Answer 11

Yes I do

Answer 12

so do it. write what u think the answer is

Answer 13

so the right side would be 2?

Answer 14

yes.
and the left side?

Answer 15

2(1)=2?

Answer 16

yes.

Answer 17

so
\[ \large P(1): 2=2 \]

Answer 18

what about P(k) ??

Answer 19

What would that be?

Answer 20

it means, replace EVERY n by k

Answer 21

So it would be P(k)= P(1)?

Answer 22

no. in the original problem, just replace every \(n\) with a \(k\).

Answer 23

oh so then after that, we do Pk+1= P(1+1)?

Answer 24

no. it is quite the same P(k+1) means "replace every \(n\) with \(k+1\). in the original problem.

Answer 25

so. for instance
\[ \large P(k):2+6+14+\dots+2k=2(k+1)-2 \]

Answer 26

try writting P(k+1)

Answer 27

2(k+1)? I'm not quite sure

Answer 28

no. the whole thing. look at what i wrote for P(k) !!

Answer 29

P(k+1)=2 + 6 + 14 + . . . + 2k = 2(k+1) - 2?

Answer 30

no. every n has to be replaced by k+1. so for instance, if u had 7n+3 it would then become 7(k+1)+3
got it?

Answer 31

ohh yeah okay. So what would I basically write for the answer though?

Answer 32

well
\[ \large P(k+1):2+6+14+\dots+2(k+1)=2[(k+1)+1]-2 \]

Answer 33

got it?

Answer 34

And that would be it?

Answer 35

yes.

Answer 36

Great :) thank you so much again! I really appreciate it

Answer 37

u r very welcome.