Question

WILL GIVE BEST RESPONSE
Use mathematical induction to prove the statement is true for all positive integers n.

10 + 30 + 60 + ... + 10n = 5n(n + 1)

Answer 1

step 1, is it true for n=1?

Answer 2

if so, assume its true for n=k

and then add k+1 to the mix and see if it simplifies to an n=k+1 model

Answer 3

can you show me the process please?

Answer 4

step 1: does 10(1) = 5(1)(1+1) ?

Answer 5

yes it does

Answer 6

then we have a basis step, so lets assume its true for any n=k and write

10 + 30+ ... + 10k = 5k(k + 1)

agreed?

Answer 7

yes I agree

Answer 8

now, the next step lets us know that if we add (k+1), then by induction its proved

Answer 9

lets add (k+1), but i might have said that wrong, lets add one more element to the left side; and add the same amount to the right side


10 + 30+ ... + 10k + 10(k+1) = 5k(k + 1) + 10(k+1)

Answer 10

okay I understand that part

Answer 11

im trying to make sure i understand it too :)

now, if we can get the right side to equal: 5(k+1)(k+1+1) its proved

Answer 12

\[5k(k + 1) + 10(k+1)\]factor out a (k+1)
\[(k+1)(5k+10)\]factor our a 5
\[5(k+1)(k+2)\]and rewrite k+2 as k=1+1
\[5(k+1)((k+1)+1)\]
tada!!

Answer 13

now, we know that it works for k=1

and therefore we can be assured that it works for k=2, k=3, k=4, ...

Answer 14

Thank you so much! You made it seem so much easier to me, I really appreciate it!

Answer 15

youre welcome, i had to dig deep to remember that one :)