Given: t1, t2... are real numbers such that t1 t2 ... tn = 2n2 9n 13, for every positive integer n ≥ 2. tk = 103.

To find: k

Approach:
We can use the given information to create a relation between tk and tk-1. Once we have the relation, we can use it to find tk, and then keep going back until we find t1.

- Find the relation between tk and tk-1
We know that t1 t2 ... tn = 2n2 9n 13 for every positive integer n ≥ 2.
So, t1 * t2 * ... * tk-1 * tk * tk+1 * ... * tn = 2n2 9n 13 for every positive integer n ≥ k+1.

- Simplify the expression
We can simplify the expression by dividing both sides by t1 * t2 * ... * tk-1 * tk+1 * ... * tn. This gives:
tk = (2k+1 * 9k+2 * 13) / (t1 * t2 * ... * tk-1 * tk+1 * ... * tn)

- Substitute the value of tk-1 in the above expression
We know that tk-1 = (2k * 9k+1 * 13) / (t1 * t2 * ... * tk-2 * tk * tk+1 * ... * tn)
Substituting this value in the expression for tk, we get:
tk = (2k+1 * 9k+2 * 13 * t1 * t2 * ... * tk-2 * tk+1 * ... * tn) / (2k * 9k+1 * 13 * t1 * t2 * ... * tk-2 * tk+1 * ... * tn)
Simplifying this expression, we get:
tk = (2k+1 * 9k+1) / (2k * 9k)

- Substitute the value of tk-1 in the above expression
We can keep substituting the value of tk-1 in the above expression until we get t1. Let's do it for k = 4:
t4 = (2*4+1 * 9*4+1) / (2*4 * 9*4) * t3
t3 = (2*3+1 * 9*3+1) / (2*3 * 9*3) * t2
t2 = (2*2+1 * 9*2+1) / (2*2 * 9*2) * t1

- Substitute the given value of tk
We know that tk = 103. Substituting this value in the expression for t4, we get:
103 = (2*4+1 * 9*4+1) / (2*4 * 9*4) * t3
t3 = 103 * (2*4 * 9*4) / (2*4+1 * 9*4+1)
t3 = 103 * 1296 / 4104
t3 = 32.5

- Keep substituting to