Divide both members by r- 1, FORMULA (2), In this formula the ratio is raised to the power of the number of terms, which may be designated by n, and the formula will become, S= a (No6 — 1) S= (3) In this formula, substitute the formula of last term; thus, a ( −1) ↑ - 1 nth term or 1 = ar2-1. Multiply both members by r, rl = arr. nl (4) COR.-The sum of the series is equal to the product of the ratio and last term, diminished by the first term and divided by the ratio less one. = EXAMPLES. = = 1. Find the 10th term in the series, 1, 2, 4, 8, 16, etc. FORMULA. nth term ar”-1 = 1× 29 = 1×2×2×2×2×2×2×2×2×2 = 512. 2. Find the 6th term of 64, 32, 16, etc. The ratio is 1. nth term ɑrn-1. = × 6th 64x (4)3 = 64 × 1 × 1 × 1 × 1 × 1 = 2. 3. Find the sum of 5 terms of 1, 2, 4, 8, etc. a (pn - 1) S= r 1 4. Find the sum of 8 terms of 1, 3, 9, 27, etc. 1 (251) = a 1 | (31) = 31. 1 COMPOUND INTEREST. Compound Interest consists in adding the interest to the principal as often as the interest becomes due, until the end of the time at which it is at interest. When a sum of money is at interest for a considerable time, and no interest has been collected, although it was specified that the interest was payable annually, semiannually, or quarterly, and the interest is computed on the principal for the specified time of interest, and at the end of such period added to the principal, this is called Compound Interest; thus, the interest of $1 at 6% for 1 year is $.06, which if added to the principal makes $1.06. The ratio will be 188, or 1.06; therefore, 1 $1.06 is the amt. of $1 for 1 yr. at 6% 1.06 6 yrs. at 6% = R. = R2. = R3. = R1. = R5. = R6. .418519, called 42 cents, is the compound interest of $1 for 6 years at 6%. Let R = 1.06, then R2, etc. will be the amt. of $1 for one, two, etc. years, R" for n years. If the interest were payable semi-annually, it would be 1.03 to the power expressed by double the number of years; thus, 1.03 R. (1.03) 12 R12 If quarterly, (1.015)24 =R24 R representing respectively 1.06, 1.03, and 1.015. After the manner of the above computation, form a table for the amount of $1 for 50 years at 2%, 21%, etc., to 10%, and as a number in being multiplied by itself is said to be raised to higher powers, as above, RX R = R2, RX RX R = R3, R2 x R3 R5, R4 x R3 R'. When a number is raised to higher powers, the product has the sum of the powers; therefore, by this table, the compound amount of $1 for any number of years can easily be found, and from the amount subtract 1 and the remainder will be the compound interest. REM.—If there be a few months or days time above the specified time which cannot be had from the tables, take the highest amount in the tables and compute the balance of time on it as in simple interest, and add it to the amount obtained from the tables. Yrs. TABLE showing the amount of $1, at 21, 3, 31, 4, 5, and 6%, compound int., from 1 to 20 years. 21%. 3%. 31%. 4%. 1 1.025000 1.030000 1.035000 1.040000 1.050000 1.060000 2 1.050625 1.060900 1.071225 1.081600 1.102500 1.123600 3 1.076891 1.092727 1.108718 1.124864 1.157625 1.191016 4 1.103813 1.125509 1.147523 1.169859 1.215506 1.262477 5 1.131408 1.159274 1.187686 1.216653 1.2762821.338226 5%. EXAMPLES. 6%. 6 1.159693 1.194052 1.229255 1.265319 1.340096 1.418519 7 1.188686 1.229874 1.272279 1.315932 1.407100 1.503630 81.218403 1.266770 1.316809 1.368569 1.477455 1.593848 9 1.248863 1.304773 1.362897 1.423312 1.551328 1.689479 10 1.280085 1.343916 1.410599 1.480244 1.628885 1.790848 11 1.312087 1.384234 1.459970 1.539454 1.710339 1.898299 12 1.344889 1.425761 1.511069 1.601032 1.795856 2.012197 1.3785111.468534 1.563956 1.665074 1.885649 2.132928 13 14 1.412974 1.512590 1.618695 1.731676 1.979932 2.260904 15 1.448298 1.557967 1.675349 1.800944 2.078928 2.396558 16 1.484506 1.604706 1.733986 1.872981 2.182875 2.540352 17 1.5216181.652848 1.794676 1.947901 2.292018 2.692773 181.559659 1.702433 1.857489 2.025817 2.406619 2.854339 19 1.5986501.753506 1.922501 2.106849 2.526950 3.025600 201.638616 1.806111 1.989789 2.191123 2.653298 3.207136 1. What is the compound amount of $500 for 12 yrs., at 6% ? The number in the table opposite 12 years, at 6%, is 2.0121965 × 500 = $1006.09825. If for 12 years and 6 months, 2.0121965 x 1.03 2.0725624 × 500 = $1036.2812. If for 12 years and 3 months, × 1.015. For any other time, as for a few days, compute the interest on the tabular number as you would in simple interest, and add it to the number of the table. 2. Find the compound amount of $100 for 100 years, at 6%. The tabular number for 50 years is 18.420154 18.420154 339.3021 × 100 = $33930.21. R50 × R50 = R100 3. Find the compound amount of $100 for 5 yr. 1 mo. 20 da., at 6%. The tabular amount for 5 years is 1.3382256. 1 mo. 20 da. = 50 da. 6500 = 680 = Tho• 1.3382256 × TRO = .0111519. 1.3382256 .0111519 1.3493775 x 100 = $134.94, amt. for 5 yr. 1 mo. 20 da. In the above, the interest was payable annually. 4. Find the amount of $500 for 5 years, at 6%, interest payable semi-annually. The tabular number opposite 3% for 10 years is 1.34391638 500 $671.95819 |