Determine the total number of
individual rational terms in the
multinomial expansion of each of the following expressions:
(I) (√12 + √32 + √48 + √50)
6
(II) (√12 + √32 + √48 + √50)
8
Hence or otherwise, determine the total number of individual rational terms in the multinomial expansion of (√12 + √32 + √48 + √50)
2n in terms of n, whenever n is a positive integer.
Using a formula from the link, the total number of terms in this expanded multinomial is (2n+3)*(2n+2)*(2n+1)/6. This is also the formula for the (2n+1)th tetrahedral number.
The integer terms will occur when the sum of the exponents for sqrt(12) and sqrt(48) is even and when the sum of the exponents for sqrt(32) and sqrt(50). These two conditions are equivalent since the exponent 2n is always even.
Another way to sum the value of the (2n+1)th tetrahedral number is Sum {k=1 to 2n+1} k*(2n+2-k). This is also reflected in the multinomial. The kth term of this sum count the number of terms of the multinomial have the sum of the exponents for sqrt(12) and sqrt(48) equal to k-1.
Let 2j-1 = k. Then the number of integer terms will equal Sum {j=1 to n+1} (2j-1)*(2n+3-2j). This sum equals the
octahedral numbers, whose explicit formula is t(x)=(2x^3+x)/3. For this problem, x=n+1.
Then the number of integer terms in the multinomial expansion of (sqrt(12)+sqrt(32)+sqrt(48)+sqrt(50))^(2n) equals (2(n+1)^3+(n+1))/3.
Solution
