Question

Medal and fan !!!!Please help I don't get this its about mathematical induction.
use mathematical induction to prove that the statement holds for all positive integers. 2+4+6+...+2n=n^2+n

can you display all the steps involved with an explanation of how you get the answer because I really want to understand how to do this. thank you.

Answer 1

first show that its true for n = k
then its true forn = k + 1

its obviously true for n = 1

so if you can prove the first 2 above then its must be true for n = 1,2, 3 and so on

Answer 2

how would I do that? like is there an equation I could use ( sums of powers of integers equations?????)

Answer 3

you also need to find the formula for the nth term of this series in order to do the above

Answer 4

what do you think the nth term is?

Answer 5

2?

Answer 6

lwhat is the relationship between n and nth term

n nth term

1 1
2 4
3 6
4 8


can you see the pattern?

Answer 7

its 2n right?

Answer 8

first tern is 2 @cwrw238

Answer 9

what will I do with the 2n then?

Answer 10

ok
assume that the sum formula is true for n = k

then sum of k terms = k^2 + k

so the sum of k+1 terms ( which is sum of k terms + (k + 1) th term )
= k^2 + k + 2(k+1)

agreed?

Answer 11

In my first post i should have said ASSUME true for k not show true for k

Answer 12

do you follow the argument so far?

Answer 13

nth term = 2n so (k+1)th term = 2(k +1)

Answer 14

somewhat

Answer 15

ok
so so far we have worked out that
sum of (k+1) terms = k^2 + k + 2(k + 1)
k^2 + k + 2(k + 1)
= k^2 + k + 2k + 2
=k^2 + 2k + 1 + k + 1
= (k + 1)^2 + k + 1

Answer 16

- notice that this sum formula is the same as the formula for the of k terms except that k is replaced by k+1

ok?

Answer 17

* same as sum for k terms except that k is replaced by (k +1)

Answer 18

ah well!! - he seems to have given up!

Answer 19

good you are back

do you follow this ok?

Answer 20

wait who? im lost. so k is replaced by k+1 what happens next to the equation?
oh its a she.

Answer 21

yep im a she lol

Answer 22

so am I lol im terrible at math lol

Answer 23

if you compare the 2 formula
sum of k terms = k^2 + k
sum of k+1 terms = (k+1)^2+ k + 1

- from this we can say that if the sum of k formula is true then the sum of (k+1) formula is true

right?

Answer 24

LOL no one is terrible at math its this thought which makes them think they are terrible keep practicing no one is as bad as they think they are.

Answer 25

i agree

Answer 26

yeah k+1 is true. when it comes to calculus I am alright I have to admit but at times I get dumbfounded with these problems because I guess what they ask confuses me

Answer 27

thats the algorithm for proving induction problems you should probably learn it.

Answer 28

ik that in order to prove the induction I need to make sure pi is true and the truth of p implies the truth of pk+1 for every positive k , then pn has to be true for all positive integers. I get that to a degree when it comes the expression itself I just get confused with the steps involved.

Answer 29

yea Proof by Induction can be confusing at first basically for a sequence it consists of

1 Assume that the sum formula is true for n = k

2 Work out what the nth term is. Then put n = k+1 and add this to the formula for sum formula for n = k

3. Then manipulate this algebraically to try and get the same formula as k except that k+1 replaces k. This proves that if the sum formula is true for k then its true for k+1.

4. Now show that sum formula is true for n = 1.

5. This concludes the proof because its true for k+1 if sum of k is correct and its true for k=1 then its must be true for k=2 and k = 3 and so one. So true for all n.

Answer 30

since we found k+1 to be true there is no more to do in the equation then?

Answer 31

- yes - you have to contrive to get the expression in same form as k expression can be tricky

Answer 32

except to show that its true for k =1 which it obviously is in this case. Sum of 2 = 2!

Answer 33

so sum of 3 and 4 would be 3, 4?

Answer 34

no want i meant was sum of 1 term = 2 as 2 is first term

sum of 2 terms = 2+4

Answer 35

oh...hehe.

Answer 36

lol

Answer 37

test the formula if you like
sum of 2 terms = 2^2 + 2 = 6

2 +4 = 6

Answer 38

okay so the next thing to do would be to add 6+___= to get k=3?

Answer 39

right

Answer 40

but you would not do this if you were asked to find sum of 90 terms for instance - it would take a lot of adding

you would replace n by 90 in the formula

sum of 90 terms = 90^2 + 90 = 8190

Answer 41

ok so it would be 6+ 8 or 6+12?

Answer 42

you mean the sum of 3 terms?

Answer 43

yes

Answer 44

sorry 2 + 4 +6 = 12

Answer 45

k=4 would then be 2+4+6+12=?

Answer 46

the important thing is for you to try and understand the method of induction and practice some problems

Answer 47

no the next term after 6 is 8 - they are going up in 2's

Answer 48

ok

Answer 49

thanks for the incredible help!

Answer 50

yw