Please help me I can't solve questions of mathematical induction.I tried a lot but I always get confused in the (k+1)th step.Please tell what to do?
u take a solved example and try 2 understand these steps
u first take n=1....i
then let p(k) is true .....ii
and then take p(k+1).....iii
in this u need to copy lhs of ii and write an extra term with "n " of last term replaced with "k+1" and in rhs simply replace n with k+1
after that take lhs of iii
then replace entire lhs with rhs of ii + term with k+1
then try to take common
ultimately u will get the desired rhs
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step 1 is usually easy, we just have to prove it is true for n=1
Step 2 is best done this way:
- Assume it is true for n=k
- Prove it is true for n=k+1 (we can use the n=k case as a fact.)
Example: is 3n−1 a multiple of 2?
Is that true? Let us find out.
1. Show it is true for n=1
31−1 = 3−1 = 2
Yes 2 is a multiple of 2. That was easy.
31−1 is true
2. Assume it is true for n=k
3k−1 is true
(Hang on! How do we know that? We don't!
It is an assumption ... that we treat
as a fact for the rest of this example)
Now, prove that 3k+1−1 is a multiple of 2
3k+1 is also 3×3k
And then split 3× into 2× and 1×
And each of these are multiples of 2
- 2×3k is a multiple of 2 (we are multiplying by 2)
- 3k−1 is true (we said that in the assumption above)
3k+1−1 is true