To find the final number of terms of a binomial (a polynomial of two terms such as: A+B) raised to any power “e” the answer is to simply add one to the exponent of the polynomial.
T=E+1 where T=Final Number of Terms
E=The Exponent.
This simple fact is taught to all students of Algebra very early in the study of polynomials.
What is not taught is the ”simple” expression to find the final number of terms of say
(A+B+C) raised to any power.
To find the final number of terms of a trinomial raised to any power “e”, many expansions of
(A+B+C)e were carried out.
The following data was obtained:
Exponent
Number of Terms
Difference
Sum of Number of Terms
Square Root
2 6
3 10 4 16 4
4 15 5 25 5
5 21 6 36 6
6 28 7 49 7
7 36 8 64 8
8 45 9 81 9
9 55 10 100 10
10 66 11 121 11
This arrangement, rectangular in form, has several interesting properties:
1- The sum of consecutive number of terms is a perfect square
2- The perfect squares are the squares of the difference of the two terms that were added
Number of Terms
Difference
Sum of Number of Terms
Square Root
2 6
3 10 4 16 4
4 15 5 25 5
5 21 6 36 6
6 28 7 49 7
7 36 8 64 8
8 45 9 81 9
9 55 10 100 10
10 66 11 121 11
This arrangement, rectangular in form, has several interesting properties:
1- The sum of consecutive number of terms is a perfect square
2- The perfect squares are the squares of the difference of the two terms that were added
to obtain the square.
X+Y=A2
And
Y-X=A
X+Y=A2
And
Y-X=A
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