We are given an integer n and n-th term in a series as expressed below:
Tn = n2 - (n-1)2
We need to find Sn mod (109 + 7), where Sn is the sum of all of the terms of the given series and,
Sn = T1 + T2 + T3 + T4 + ...... + Tn
Examples:
Input : 229137999 Output : 218194447 Input : 344936985 Output : 788019571
Let us do some calculations, before writing the program. Tn can be reduced to give 2n-1 . Let’s see how:
Given, Tn = n2 - (n-1)2 Or, Tn = n2 - (1 + n2 - 2n) Or, Tn = n2 - 1 - n2 + 2n Or, Tn = 2n - 1.
Now, we need to find ?Tn.
?Tn = ?(2n – 1)
We can simplify the above formula as,
?(2n – 1) = 2*?n – ?1
Or, ?(2n – 1) = 2*?n – n.
Where, ?n is the sum of first n natural numbers.
We know the sum of n natural number = n(n+1)/2.
Therefore, putting this value in the above equation we will get,
?Tn = (2*(n)*(n+1)/2)-n = n2
Now the value of n2 can be very large. So instead of direct squaring n and taking mod of the result. We will use the property of modular multiplication for calculating squares:
(a*b)%k = ((a%k)*(b%k))%k
// CPP program to find sum of given // series. #include <bits/stdc++.h> using namespace std;
#define mod 1000000007 // Function to find sum of series // with nth term as n^2 - (n-1)^2 long long findSum( long long n)
{ return ((n % mod) * (n % mod)) % mod;
} // Driver code int main()
{ long long n = 229137999;
cout << findSum(n);
return 0;
} |
218194447
Please refer complete article on Find sum of Series with n-th term as n^2 – (n-1)^2 for more details!