# Proving First Order Recursive Formula (1 Viewer)

#### SB257426

##### Very Important User
I have come across some questions that are asking me to prove the general formula for a certain sequence. I do now know how to tackle these questions. Can someone please tell me how to do them?

Here is a problem from the question set that appeared:

A sequence is given by the first order recursive formula:

Prove the general formula for the sequence is:

Any help will be appreciated,
Cheers

induction

#### ExtremelyBoredUser

##### Bored Uni Student
I have come across some questions that are asking me to prove the general formula for a certain sequence. I do now know how to tackle these questions. Can someone please tell me how to do them?

Here is a problem from the question set that appeared:

A sequence is given by the first order recursive formula:
View attachment 37977
Prove the general formula for the sequence is:
View attachment 37978

Any help will be appreciated,
Cheers
You can do induction but why don't you write out a_1, a_2, a_3 and see if there's a pattern you can exploit. Only saying because it seems to extremely fit the geometric series formula.

#### synthesisFR

##### Well-Known Member
You can do induction but why don't you write out a_1, a_2, a_3 and see if there's a pattern you can exploit. Only saying because it seems to extremely fit the geometric series formula.
doesn't first order recursive in the syllabus fall under proof by induction tho so u would have to prove it by induction?

#### ExtremelyBoredUser

##### Bored Uni Student
doesn't first order recursive in the syllabus fall under proof by induction tho so u would have to prove it by induction?
Unless it explicitly states "Prove ... using principle of mathematical induction" then I'd consider it fair game. Proof questions just require you to use some method of proof to demonstrate the theorem, by that logic some questions you can't use certain techniques because they're not in the topic

#### Drongoski

##### Well-Known Member
As suggested by synthesisFR by Induction:

$\bg_white a_n =2^n -1 is true for n = 1 as 2^1 - 1 = 1 = a_1\\ \\ Assume true for n = k (k \geq 1)\\ \\ i.e. a_k = 2^k - 1 \\ \\ \therefore a_{k+1} = 2a_k + 1 = 2(2^k - 1) + 1 = 2^{k+1} - 1$

So, if the formula holds for n = k, it holds also for n = k + 1
Therefore by the Principle of Mathematical Induction, the formula holds for all positive integers 'n'.

So, it's quite straightforward. Maybe you just didn't know how to proceed with this unfamiliar type of questions.

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