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?

in this step you only have to put (k+1) instead of any integer given in the question like n.

and to elliminate your confusion, **pratice**..

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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 3

^{n}−1 a multiple of 2?**Is that true? Let us find out.****1.**Show it is true for**n=1****3**^{1}−1 = 3−1 = 2Yes 2 is a multiple of 2. That was easy.

3

^{1}−1 is true**2.**Assume it is true for**n=k**3

^{k}−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

**3**is a multiple of 2^{k+1}−1**3**is also^{k+1}**3×3**^{k}And then split

**3×**into**2×**and**1×**And each of these are multiples of 2

Because:

**2×3**is a multiple of 2 (we are multiplying by 2)^{k}**3**(we said that in the assumption above)^{k}−1 is true-
So:

3

^{k+1}−1 is trueDONE!

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