logarithms, add the logarithm of the first term to the logarithm of the ratio multiplied by the number of terms less one; the sum will be the logarithm of the last term.

EXAMPLE.

1. The first term of a geometrical series is 4, the ratio 5, and the number of terms 61. Required the last term.

Or, log. = log. 4+(log. 5)×60=0.60205999+41.9382000

=42.54025999.

Hence, finding the natural number corresponding with 42.54025999,

7=3469479392577934009746744427570344331708876. 2. The formula for the sum of the terms in a geometrical series is (Art. 386)

S_arn—a
-1

In this formula, if n is not a small number, it will be found convenient to find the value of ar" by taking

Log. (ar")= log. a+(log. r)×n.

Thus, we may find the value of ar" in the same way that we found the last term in the preceding case.

3. To obtain a formula for the number of terms, let us resume the formula for the sum of the terms,

1. The sum of a geometrical series is 6560, the first term 2, and the ratio 3. Required the number of terms.

Here

S=6560, a=2, and r=3;

2. The sum of a geometrical series is 1023, the first term 1, and the ratio 2. Required the number of terms.

Ans. 10.

3. The sum of a geometrical series is 640, the first term 4, and the ratio 1.01. Required the number of terms.

V. COMPOUND INTEREST.

410. By means of logarithms we may also determine the number of years it will take a given principal, at a given rate, compound interest, to gain a certain amount. Thus, in Art. 219, we have the formula

A= P(1+r)" ;

Taking the logarithms, log. A= log. P+(log. (1+r))×n; Transposing, (log. (1+r))×n= log. A— log. P;

log. A-log. P n= log. (1+r)

Note.-By means of this formula we may ascertain the number of years it would take a sum of money to double, triple, &c., or amount to m times itself, when put out at compound interest, at a given rate per cent.

EXAMPLES.

1. A man loans $1250, at 6 per cent. compound interest. In what time will it amount to $4008.92 ?

In this example, A=$4008.92, P=1250, and r+1=,06+

2. At 6 per cent. compound interest, in how many years

will $1200 amount to $2149.19 ?

·M M

Ans. 10 years.

3. At 6 per cent. compound interest, in how many years will money amount to double, triple, and quadruple the original sum? Ans. 11,9955, 18,8145, and 23,791 years.

APPENDIX.

THE three following sections, as they contain those portions of algebraic analysis which are seldom pursued in academies and schools, but which are nevertheless essential to the successful prosecution of the higher branches of the mathematical course, have been thrown into the form of an appendix.