Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

the number of terms is finite, the sum of the series added to
unity, is equal to the double of the last term. Thus the sum of
1 +2+1 is the double of 2. The sum of 1+2+4+1 is the
double of 4, and so of the rest.
In the geometrical series 1+3+9+27+81, &c. the triple
of the last term exceeds the double of the sum of the series by
unity. Thus in two terms the triple of 3, i.e. 9, exceeds the
double of the sum l+3, i.e. 4, by 1. Thus the triple of 9,
i.e. 27, exceeds the double of the sum, 1+3+9, i.e. 13 by I,
and so of the rest.
In the geometrical series 1+4+16+64+256, &c. the quadruple
of the last term exceeds the triple of the sum of the
series by unity.
And in the geometrical series 1 +5+25+ 125+625, kc. the
quintuple of the last term exceeds the quadruple of the sum of
the series by unity.
Again, in the series 1 +2+4+8+ 16, &c. when the number
of terms is finite, the last term added to the last term less by
unity, is equal to the sum of the series. Thus 2 added to 2 less
by 1=1+2. 4 added to 4 less by 3=1+2+4, &c.
In the series 1+3+9+27+81,
&c. if unity is subtracted
from the last term, and the remainder divided by 2 is added to
the last term, the sum is equal to the sum of the series. Thus
3--1=2 this divided by 2=1 and 1+3=4= the sum of the
two first terms; 9-1=8 and 8 divided by 2=4, and 4+9
=13=1+3+9, &c.
In the series 1+4+16+64+256, &c. if unity is subtracted
from the last term, and the remainder divided by 3 is added to
the last term, the sum is equal to the sum of the series.
In the series 1+5+25+ 125+625, &c. if unity is subtracted
from the last term, and the remainder divided by 4 is added to
the last term, the sum is equal to the sum of the series.