# CKVIRS - EDITORIAL

Author: Pranjul Pal
Tester: Deependra
Editorialist: Pranjul Pal

MEDIUM

# PREREQUISITES:

Matrix Exponentiation

# PROBLEM:

Calculate the total no. of infected people till N^{th} day \bmod 1000000007.
Given, every infected person till i^{th} day will infect x different non-infected persons on (i+1)^{th} day and y different non-infected persons on (i+2)^{th} day.

# QUICK EXPLANATION:

Let’s call F(d) = total no.of infected persons till d^{th} day.
It can be proved that the total no. of infected persons till N^{th} day:
i.e. F(N)=(x+1)*F(N-1)+y*F(N-2)

# EXPLANATION:

Let’s call F(d) = total no.of infected persons till d^{th} day.
It is given that spread of the virus started with 1 person i.e. F(1)=1 .
Since, this 1^{st} person with infect x non-infected persons on 2^{nd} day and y non-infected persons on 3^{rd} day.
So, total infected persons on 2^{nd} day i.e. F(2) = x*F(1)+F(1)
Let d be any day (d>=3):
Total no. infected persons till (d-1)^{th} day are F(d-1).
And total no. infected persons till (d-2)^{th} day are F(d-2).
So, F(d) can be calculated as F(d)=x*F(d-1)+y*F(d-2) +F(d-1).
Where x*F(d-1)+y*F(d-2) are the total no. of persons who are newly infected on d^{th} day and F(d-1) are already infected people till (d-1)^{th} day.
Obtained Formula: F(N)=(x+1)*F(N-1)+y*F(N-2)
Since the values are large, we need to calculate F(N) \bmod 1000000007 using Matrix Exponentiation .

# SOLUTIONS:

Setter's Solution
``````#include <bits/stdc++.h>
using namespace std;
#define ll long long
#define ln "\n"
#define pb push_back
#define pll pair<ll,ll>
#define ppll pair<ll,pll>
#define vll vector<ll>
#define vpll vector<pll>
#define vvll vector<vector<ll>>
#define sll stack<ll>
#define qll queue<ll>
#define mp make_pair
#define f first
#define s second
#define bs binary_search
#define lb lower_bound
#define ub upper_bound
#define Test ll t;cin>>t; while(t--)
#define fast_io ios_base::sync_with_stdio(false);cin.tie(NULL);
#define init(x,n,v) for(ll i=0;i<=n;i++) x[i]=v;
#define all(x) x.begin(),x.end()
#define pi 3.14159265358979323846
ll MOD = 1e9+7 , MAX = 1e18;
vvll mat;
void mul(vvll &a,vvll &b)
{
ll i,j,k;
vvll c={{0,0},{0,0}};
for(i=0;i<2;i++)
{
for(j=0;j<2;j++)
{
for(k=0;k<2;k++)
{
c[i][j]=(c[i][j]+(a[i][k]*b[k][j]%MOD))%MOD;
}
}
}
a=c;
}
void power(ll n)
{
vvll res={{1,0},{0,1}};
while(n)
{
if(n&1) mul(res,mat);
mul(mat,mat);
n/=2;
}
mat=res;
}
int main()
{
#ifndef ONLINE_JUDGE
freopen("input1.txt","r",stdin);
freopen("output1.txt","w",stdout);
#endif
fast_io;
Test
{
ll x,y,n,ans;
cin>>x>>y>>n;
if(n==1) ans=1;
else if(n==2) ans=x+1;
else
{
mat={{0,y},{1,x+1}};
power(n-2);
ans=(mat[0][1]+((x+1)*mat[1][1])%MOD)%MOD;
}
cout<<ans<<ln;
}
return 0;
}
``````
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