For an integer n≥1 we consider sequences of 2n numbers, each equal to 0,−1 or 1. The sum product value of such a sequence is calculated by first multiplying each pair of numbers from the sequence, and then adding all the results together.
For example, if we take n=2 and the sequence 0,1,1,−1, then we find the products 0⋅1,0⋅1,0⋅−1,1⋅1,1⋅−1,1⋅−1. Adding these six results gives the sum product value of this sequence: 0+0+0+1+(−1)+(−1)=−1. The sum product value of this sequence is therefore smaller than the sum product value of the sequence 0,0,0,0, which equals 0.
Determine for each integer n≥1 the smallest sum product value that such a sequence of 2n numbers could have.Attention: you are required to prove that a smaller sum product value is impossible. algebraSequenceSumProductminimum