Given three integers A, X, and n. The task is to find the middle term in the binomial expansion series.
Examples:
Input : A = 1, X = 1, n = 6
Output : MiddleTerm = 20Input : A = 2, X = 4, n = 7
Output : MiddleTerm1 = 35840, MiddleTerm2 = 71680
Approach
(A + X)n = nC0 An X0 + nC1 An-1 X1 + nC2 An-2 X2 + ……… + nCn-1 A1 Xn-1 + nCn A0 Xn
Total number of term in the binomial expansion of (A + X)n is (n + 1).
General term in binomial expansion is given by:
Tr+1 = nCr An-r Xr
If n is even number:
Let m be the middle term of binomial expansion series, then
n = 2m
m = n / 2
We know that there will be n + 1 term so,
n + 1 = 2m +1
In this case, there will is only one middle term. This middle term is (m + 1)th term.
Hence, the middle term
Tm+1 = nCmAn-mXm
if n is odd number:
Let m be the middle term of binomial expansion series, then
let n = 2m + 1
m = (n-1) / 2
number of terms = n + 1 = 2m + 1 + 1 = 2m + 2
In this case there will be two middle terms. These middle terms will be (m + 1)th and (m + 2)th term.
Hence, the middle terms are :
Tm+1 = nC(n-1)/2 A(n+1)/2 X(n-1)/2
Tm+2 = nC(n+1)/2 A(n-1)/2 X(n+1)/2
// C++ program to find the middle term // in binomial expansion series. #include <bits/stdc++.h> using namespace std;
// function to calculate // factorial of a number int factorial( int n)
{ int fact = 1;
for ( int i = 1; i <= n; i++)
fact *= i;
return fact;
} // Function to find middle term in // binomial expansion series. void findMiddleTerm( int A, int X, int n)
{ int i, j, aPow, xPow;
float middleTerm1, middleTerm2;
if (n % 2 == 0)
{
// If n is even
// calculating the middle term
i = n / 2;
// calculating the value of A to
// the power k and X to the power k
aPow = ( int ) pow (A, n - i);
xPow = ( int ) pow (X, i);
middleTerm1 = (( float )factorial(n) /
(factorial(n - i) * factorial(i)))
* aPow * xPow;
cout << "MiddleTerm = "
<< middleTerm1 << endl;
}
else {
// If n is odd
// calculating the middle term
i = (n - 1) / 2;
j = (n + 1) / 2;
// calculating the value of A to the
// power k and X to the power k
aPow = ( int ) pow (A, n - i);
xPow = ( int ) pow (X, i);
middleTerm1 = (( float )factorial(n) /
(factorial(n - i) * factorial(i)))
* aPow * xPow;
// calculating the value of A to the
// power k and X to the power k
aPow = ( int ) pow (A, n - j);
xPow = ( int ) pow (X, j);
middleTerm2 = (( float )factorial(n) /
(factorial(n - j) * factorial(j)))
* aPow * xPow;
cout << "MiddleTerm1 = "
<< middleTerm1 << endl;
cout << "MiddleTerm2 = "
<< middleTerm2 << endl;
}
} // Driver code int main()
{ int n = 5, A = 2, X = 3;
// function call
findMiddleTerm(A, X, n);
return 0;
} |
// Java program to find the middle term // in binomial expansion series. import java.math.*;
class GFG {
// function to calculate factorial
// of a number
static int factorial( int n)
{
int fact = 1 , i;
if (n == 0 )
return 1 ;
for (i = 1 ; i <= n; i++)
fact *= i;
return fact;
}
// Function to find middle term in
// binomial expansion series.
static void findmiddle( int A, int X, int n)
{
int i, j, aPow, xPow;
float middleTerm1, middleTerm2;
if (n % 2 == 0 )
{
// If n is even
// calculating the middle term
i = n / 2 ;
// calculating the value of A to
// the power k and X to the power k
aPow = ( int )Math.pow(A, n - i);
xPow = ( int )Math.pow(X, i);
middleTerm1 = (( float )factorial(n) /
(factorial(n - i) * factorial(i)))
* aPow * xPow;
System.out.println( "MiddleTerm = "
+ middleTerm1);
}
else {
// If n is odd
// calculating the middle term
i = (n - 1 ) / 2 ;
j = (n + 1 ) / 2 ;
// calculating the value of A to the
// power k and X to the power k
aPow = ( int )Math.pow(A, n - i);
xPow = ( int )Math.pow(X, i);
middleTerm1 = (( float )factorial(n) /
(factorial(n - i) * factorial(i)))
* aPow * xPow;
// calculating the value of A to the
// power k and X to the power k
aPow = ( int )Math.pow(A, n - j);
xPow = ( int )Math.pow(X, j);
middleTerm2 = (( float )factorial(n) /
(factorial(n - j) * factorial(j)))
* aPow * xPow;
System.out.println( "MiddleTerm1 = "
+ middleTerm1);
System.out.println( "MiddleTerm2 = "
+ middleTerm2);
}
}
// Driver code
public static void main(String[] args)
{
int n = 6 , A = 2 , X = 4 ;
// calling the function
findmiddle(A, X, n);
}
} |
# Python3 program to find the middle term # in binomial expansion series. import math
# function to calculate # factorial of a number def factorial(n) :
fact = 1
for i in range ( 1 , n + 1 ) :
fact = fact * i
return fact;
# Function to find middle term in # binomial expansion series. def findMiddleTerm(A, X, n) :
if (n % 2 = = 0 ) :
# If n is even
# calculating the middle term
i = int (n / 2 )
# calculating the value of A to
# the power k and X to the power k
aPow = int (math. pow (A, n - i))
xPow = int (math. pow (X, i))
middleTerm1 = ((math.factorial(n) /
(math.factorial(n - i)
* math.factorial(i)))
* aPow * xPow)
print ( "MiddleTerm = {}" .
format (middleTerm1))
else :
# If n is odd
# calculating the middle term
i = int ((n - 1 ) / 2 )
j = int ((n + 1 ) / 2 )
# calculating the value of A to the
# power k and X to the power k
aPow = int (math. pow (A, n - i))
xPow = int (math. pow (X, i))
middleTerm1 = ((math.factorial(n)
/ (math.factorial(n - i)
* math.factorial(i)))
* aPow * xPow)
# calculating the value of A to the
# power k and X to the power k
aPow = int (math. pow (A, n - j))
xPow = int (math. pow (X, j))
middleTerm2 = ((math.factorial(n)
/ (math.factorial(n - j)
* math.factorial(j)))
* aPow * xPow)
print ( "MiddleTerm1 = {}" .
format ( int (middleTerm1)))
print ( "MiddleTerm2 = {}" .
format ( int (middleTerm2)))
# Driver code n = 5
A = 2
X = 3
# function call findMiddleTerm(A, X, n) # This code is contributed by # manishshaw1. |
// C# program to find the middle term // in binomial expansion series. using System;
class GFG {
// function to calculate factorial
// of a number
static int factorial( int n)
{
int fact = 1, i;
if (n == 0)
return 1;
for (i = 1; i <= n; i++)
fact *= i;
return fact;
}
// Function to find middle term in
// binomial expansion series.
static void findmiddle( int A, int X, int n)
{
int i, j, aPow, xPow;
float middleTerm1, middleTerm2;
if (n % 2 == 0)
{
// If n is even
// calculating the middle term
i = n / 2;
// calculating the value of A to
// the power k and X to the power k
aPow = ( int )Math.Pow(A, n - i);
xPow = ( int )Math.Pow(X, i);
middleTerm1 = (( float )factorial(n) /
(factorial(n - i) * factorial(i)))
* aPow * xPow;
Console.WriteLine( "MiddleTerm = "
+ middleTerm1);
}
else {
// If n is odd
// calculating the middle term
i = (n - 1) / 2;
j = (n + 1) / 2;
// calculating the value of A to the
// power k and X to the power k
aPow = ( int )Math.Pow(A, n - i);
xPow = ( int )Math.Pow(X, i);
middleTerm1 = (( float )factorial(n) /
(factorial(n - i) * factorial(i)))
* aPow * xPow;
// calculating the value of A to the
// power k and X to the power k
aPow = ( int )Math.Pow(A, n - j);
xPow = ( int )Math.Pow(X, j);
middleTerm2 = (( float )factorial(n) /
(factorial(n - j) * factorial(j)))
* aPow * xPow;
Console.WriteLine( "MiddleTerm1 = "
+ middleTerm1);
Console.WriteLine( "MiddleTerm2 = "
+ middleTerm2);
}
}
// Driver code
public static void Main()
{
int n = 5, A = 2, X = 3;
// calling the function
findmiddle(A, X, n);
}
} // This code is contributed by anuj_67. |
<?php // PHP program to find the middle term // in binomial expansion series. // function to calculate // factorial of a number function factorial(int $n )
{ $fact = 1;
for ( $i = 1; $i <= $n ; $i ++)
$fact *= $i ;
return $fact ;
} // Function to find middle term in // binomial expansion series. function findMiddleTerm( $A , $X , $n )
{ $i ; $j ;
$aPow ; $xPow ;
$middleTerm1 ;
$middleTerm2 ;
if ( $n % 2 == 0)
{
// If n is even
// calculating the middle term
$i = $n / 2;
// calculating the value of A to
// the power k and X to the power k
$aPow = pow( $A , $n - i);
$xPow = pow( $X , $i );
$middleTerm1 = $factorial ( $n ) /
(factorial( $n - $i ) * factorial( $i ))
* $aPow * $xPow ;
echo "MiddleTerm = " , "\n" ,
$middleTerm1 ;
}
else
{
// If n is odd
// calculating the middle term
$i = ( $n - 1) / 2;
$j = ( $n + 1) / 2;
// calculating the value of A to the
// power k and X to the power k
$aPow = pow( $A , $n - $i );
$xPow = pow( $X , $i );
$middleTerm1 = ((float)factorial( $n ) /
(factorial( $n - $i ) * factorial( $i )))
* $aPow * $xPow ;
// calculating the value of A to the
// power k and X to the power k
$aPow = pow( $A , $n - $j );
$xPow = pow( $X , $j );
$middleTerm2 = factorial( $n ) /
(factorial( $n - $j ) * factorial( $j ))
* $aPow * $xPow ;
echo "MiddleTerm1 = "
, $middleTerm1 , "\n" ;
echo "MiddleTerm2 = "
, $middleTerm2 ;
}
} // Driver code
$n = 5;
$A = 2;
$X = 3;
// function call
findMiddleTerm( $A , $X , $n );
// This code is contributed by Vishal Tripathi. ?> |
<script> // JavaScript program to find the middle term // in binomial expansion series. // function to calculate // factorial of a number function factorial(n)
{ let fact = 1;
for (let i = 1; i <= n; i++)
fact *= i;
return fact;
} // Function to find middle term in // binomial expansion series. function findMiddleTerm(A, X, n)
{ let i, j, aPow, xPow;
let middleTerm1, middleTerm2;
if (n % 2 == 0)
{
// If n is even
// calculating the middle term
i = Math.floor(n / 2);
// calculating the value of A to
// the power k and X to the power k
aPow = Math.floor(Math.pow(A, n - i));
xPow = Math.floor(Math.pow(X, i));
middleTerm1 = Math.floor(factorial(n) /
(factorial(n - i) * factorial(i)))
* aPow * xPow;
document.write( "MiddleTerm = "
+ middleTerm1 + "<br>" );
}
else {
// If n is odd
// calculating the middle term
i = Math.floor((n - 1) / 2);
j = Math.floor((n + 1) / 2);
// calculating the value of A to the
// power k and X to the power k
aPow = Math.floor(Math.pow(A, n - i));
xPow = Math.floor(Math.pow(X, i));
middleTerm1 = Math.floor(factorial(n) /
(factorial(n - i) * factorial(i)))
* aPow * xPow;
// calculating the value of A to the
// power k and X to the power k
aPow = Math.floor(Math.pow(A, n - j));
xPow = Math.floor(Math.pow(X, j));
middleTerm2 = Math.floor(factorial(n) /
(factorial(n - j) * factorial(j)))
* aPow * xPow;
document.write( "MiddleTerm1 = "
+ middleTerm1 + "<br>" );
document.write( "MiddleTerm2 = "
+ middleTerm2 + "<br>" );
}
} // Driver code let n = 5, A = 2, X = 3;
// function call
findMiddleTerm(A, X, n);
// This code is contributed by Surbhi Tyagi. </script> |
MiddleTerm1 = 720 MiddleTerm2 = 1080
Time Complexity: O(n)
Auxiliary Space: O(1)