28 Mr. Joneson finding the Sum of all the Coefficients [Jury, 
For the sake of simplicity, I shall divide the proposition into 
the following cases, beginning with the easiest. 
1. To find the sum of all the coefficients in the expansion of 
a monomial a, to the power of n. 
It is manifest that a" = a x aX ax .... ton terms, and 
since a has unity for its coefficient, the sum will be 1x1 x1x1 
.... tom terms = 1"; therefore, the sum of all the coefficients 
in the expansion of a monomial is 1". 
2. To find the sum of all the coefficients in the expansion of a 
binomial a + 4, to the power of n. 
By the binomial theorem, (@ + b)"= a" + n.a*"*b+n. 
— od" * OF, gee are “S .a"~> b> + &c.; and since 6 has 
unity for its coefficient, all the powers of 6 will have unity for 
their coefficients, which consequently will not affect their sum ; 
whence they may be rejected, and the sum of all the coefficients 
n—1 
2 
in the expansion of (a + 6)" will be a* + na"-'+7. 
-a*° +n. -—.— -a"*++ &e.; but @ being a monomial, 
the sum of all the coefficients in a” is 1", 
at is gate 
eka is | sapere 
a TAas let? t8or. 
These values of a", a*—', a"~’, &c. being substituted in the 
expansion, gives 1"+ ”.1"~' +n a ol eka = 2 
- 1"~* + &c. but this series is absolutely the expansion of 
(1 + 1)" = 2"; therefore the sum of all the coefficients in the 
expansion of a binomial a + 0b, to the power of x, is 2”. 
3. To find the sum of all the coefficients in the expansion of a 
trmomial a + b + c to the power of n. 
To obtain this, puta + 6 =x; then(a + b+ c)"=(e«+c)" 
seit opts 2 n—1 n—1 ee ee n—1 n—2 
fa 92. CORR oma at OF MEW es ate 
+ Xc.; and because c has unity for its coefficient, allits powers 
may be rejected, and the sum ofall the coefficients in the expan- 
% ; < : n—l 
sion of a trinomial will be 2* +.2.2"7' +n. art xz +n. 
s gn—3 cc 
n—1 n-—2 5 : : 5 
zg wT + &e. 5 but since x is a binomial, the sum of all 
the coefficients in 2* = (a+ 6)" is 2", 
a) = (a + by*-* is = Sia 
a>? =a + b)*— 78:2" ~*, Ke. 
These different powers of 2 being substituted for a", 2*~', 
x*~* &e. gives 2° + n.2""' +n. i dtr « 
n—l n—2 
Gy tS rages 
