Understanding Summation Notation

[y35dp]

I know this should be easy and the answer will be glaringly obvious in hindsight but my brain is fried and I can't for the life of me figure this out. My problem is this I have a function as follows;

V = \sum\lambda_i,j,k h_ih_jh_k (summation over i,j,k where i,j,k = 1,2,3)

I can't work out if this is

V = \lambda_1,1,1h₁³ + \lambda_1,1,2h₁²h₂ + ... + \lambda_2,3,1h₂h₃h₁ +...

with every permutation of 1, 2 and 3, this should be simple as it is taken as a given in my problem but it is driving me insane.

 

[HallsofIvy]

Yes, what you have so far is correct. And, since there are 3 indices and each can take on 3 values, there will be 3^3= 27 terms in the sum.
It would help to have a "process" for working through them- I recommend changing the last index through 1, 2, 3, then the next, etc.- just like an odometer turns over in a car.

i,j,k= 1,1,1; 1,1,2; 1,1,3; 1,2,1;1,2,2; 1,2,3; 1,3,1; 1,3,2; 1,3,3;
will be the first 9 terms.
The next 9 will be exactly the same but with first number 2 and the last 9 with first number 3.

 

[y35dp]

Thank you, you may just have rescued my mental health