Year 12s: what will be the first way you research your future options? >>

What is my mistake in this proof by induction algebra?

I must be making a mistake in the algebra in the induction step.

Question. A sequence u1, u2, u3, ... is defined by

U1 = 7/2 and

u_{n} = \frac{1}{2}u_{n-1} + n^2

for n >= 2.

Prove by induction that

u_{n} = 2n^2 - 4n + 6 - (\frac{1}{2})^n

and for all positive integers n.

Here is my working.

LHS.

u_{1} = \frac{1}{2}(\frac{7}{2}) + 2^2 = \frac{7}{4} + \frac{16}{4} = \frac{23}{4}

RHS

u_{1} = 2 (2)^2 - 4 (2) + 6 - \frac{1}{2}^2 = \frac{23}{4}

Step 2. Assume true for n=k, prove for n = k+1.

LHS

u_{k+1} = \frac{1}{2}(2k^2 - 4k + 6 - (\frac{1}{2})^k) + k^2

k^2 - 2k + 3 - \frac{1}{2}^k + k^2

= 2k^2 - 2k + 3 - \frac{1}{2}^{k+1}

RHS

2 (k+1)^2 - 4 (k+1) + 6 - \frac{1}{2}^{k+1}

2k^2 +4k + 2 -4k - 4 + 6 - \frac{1}{2}^{k+1}

2k^2 + 4 - \frac{1}{2}^{k+1}

2 (k^2 + 2) - \frac{1}{2}^{k+1}

And lhs is not same as rhs :frown:

Reply 1

DFranklin

Your initial formula for "LHS" is not right (you need an extra +k^2 inside the bracket, and a +(k+1)^2 outside the bracket instead of +k^2).

Reply 2

poorform

Less of an issue but you need to be careful with notation. In your base case you have said

u_1=...

but before that you have said

u_n

is only defined for

n\geq 2

Reply 3

DFranklin

Original post by poorform

Less of an issue but you need to be careful with notation. In your base case you have said

u_1=...

but before that you have said

u_n

is only defined for

n\geq 2

Actually, no. u_1 is defined, and then for n > 1, u_n is defined in terms of u_{n-1}. It's completely standard notation.

Reply 4

poorform

Original post by DFranklin

Actually, no. u_1 is defined, and then for n > 1, u_n is defined in terms of u_{n-1}. It's completely standard notation.

But

\displaystyle u_1=\frac{7}{2}

not

\displaystyle \frac {23}{4}

so he should have put

u_2

(edited 9 years ago)

Reply 5

DFranklin

Original post by poorform

But

\displaystyle u_1=\frac{7}{2}

not

\displaystyle \frac {23}{4}

so he should have put

u_2

Oh OK, yes, that's wrong (but it's not notation, it's just wrong).

Reply 6

poorform

Original post by DFranklin

Oh OK, yes, that's wrong (but it's not notation, it's just wrong).

Yes I see what you mean sorry my English isn't the greatest!

Quick Reply

Latest

Articles for you

How to revise for A-level Geography exams: AQA explains what to do

How to write an excellent personal statement in 10 steps

Seven things you’ll learn at university (that have nothing to do with your degree)

Guide to A-level results day 2024

What is my mistake in this proof by induction algebra?

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you