Simple induction question (which i cannot do!!!) (1 Viewer)

[sasquatch]

Just a question...

Write a formula for the sum of the first n multiples of 4 and use the method of mathematical induction to prove your formula true.

This is what ive done so far!

S_n = 4 + 8 + 12 + 16 + ... + 4n

T₂ - T₁ = 8 - 4 = 4
T₃ - T₂ = 16 - 12 = 4

T₂ - T₁ = T₃ - T₂

Therefore series is an arithmetic series.

So this is all ive done!!

Do i try to prove that the formula above equals to n/2[2a + (n-1)d] (the sum of n terms for an arithmetic series)? Or yeah.. thats what im thinking.. but not 100% sure.

Thanks.

 

[sasquatch]

I cant go on until someone answers this question... guess everyone is studying for their half yearlies.......I DID MINE TWO WEEKS AGO!!!!!!!!!!

 

[insert-username]

You've only been waiting an hour. Be patient, and people will help when they can.

Write a formula for the sum of the first n multiples of 4

I'm going to use sigma notation for this. I think I've gotten the notation right - if I haven't, please tell.

₁Σⁿ 4n = n/2[8 + 4(n-1)] (you have to equate the sum to the formula for the sum of an arithmetic series, substituting in values as you can).

use the method of mathematical induction to prove your formula true.

Step 1: Prove true for n = 1. Pretty easy.


Step 2. Assume true for n = k:

₁Σⁿ 4k = k/2[8 + 4(k-1)]


Step 3: Prove true for n = k+1

₁Σ^k+1 4(k+1) = (k+1)/2[8 + 4k]

LHS: ₁Σ^k 4k + 4(k+1)

= k/2[8 + 4(k-1)] + 4k + 4 (from assumption)

= 4k + 2k(k-1) + 4k + 4

= 2k² + 6k + 4

RHS: 4(k+1) + 2k(k+1)

= 4k + 4 + 2k² + 2k

= 2k² + 6k + 4

= LHS

Therefore, since true for n=1, and true for n=k+1 if true for n=k, is true for all n≥1 by mathematical induction.

 

[sasquatch]

erm i understood to use induction.. and i can use induction.. all i was asking was if i was to try and prove taht the formula that i provided (which is exactly the same as the formula you represeneted by sigma notion) equals to the sum of an arithemetic series.

Thanks anyway.... and i wasnt angry or anything... think you misinterperated the "emotion" in my post... erm yeah..thanks

 

[insert-username]

erm i understood to use induction.. and i can use induction.. all i was asking was if i was to try and prove taht the formula that i provided (which is exactly the same as the formula you represeneted by sigma notion) equals to the sum of an arithemetic series.

Thanks anyway.... and i wasnt angry or anything... think you misinterperated the "emotion" in my post... erm yeah..thanks

Sorry about that. At least I answered your question.


I_F