#include <bits/stdc++.h>
using namespace std;
#define ll long long int
int main() {
ll t, n;
cin>>t;
while(t--){
cin>>n;
ll i,j,err=-1, a[n], b[n-1],p;
for(i=0;i<n;i++)
cin>>a[i];
for(i=1;i<n-1;i++){
if(a[i]!= (a[i-1]+a[i+1])/2)
{
err = i;
break;
}
}
ll r[]={err,err-1,err+1};
if(err ==-1){
r[0]=0;r[1] = (n-1)/2;r[2] = n-1;
}
j=0;
ll min = INT_MAX;
while(j<3){
p =0;
bool flag= true;
for(i=0;i<n;i++){
if(i==r[j])
continue;
b[p]=a[i];
p++;
}
for(i=1;i<n-2;i++){
if(b[i]!= (b[i-1]+b[i+1])/2)
flag=false;
}
if(flag==true && min>a[r[j]])
min = a[r[j]];
j++;
}
if(min== INT_MAX){
cout<<"-1\n";
}
else{
cout<<min<<"\n";
}
}
return 0;
}
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