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453. To find the number of terms when the extremes and common difference are given.
1. The extremes are 2 and 23, and the common difference 2. Find the number of terms.
7+18, No. Terms.
Examining the series, 2, 5, 8, 11, 14, analyzed in 451, we see that after taking away the first term from any term we have left the common difference taken as many times as the number of terms less 1.
Hence we take away 2, the first term, from 23, the last term, and the remainder 21 is 7 times 3. Since 7 is the number of terms less 1 the number is 7+ 1 = 8.
RULE. - Divide the difference of the extremes by the common difference, and add 1 to the quotient.
2. The extremes are 7 and 43, and the common difference is 4. What is the number of terms? Ans. 10. 3. The first term is 2, the last term is 40, and the common difference is 71. What is the number of terms? Ans. 6.
4. A laborer agreed to build a fence on the following conditions: for the first rod he was to have 6 cents, with an increase of 4 cents on each successive rod; the last rod came to 226 cents. How many rods did he build?
Ans. 56 rods.
454. To find the sum of all the terms when the extremes and number of terms are given.
1. The extremes are 2 and 14 and the number of terms 5. What is the sum of the series?
16+16 +16+ 16 + 16 = 80, twice the sum.
80 ÷ 2 = 40, the sum.
SOLUTION. To deduce a rule for finding the sum of all the terms, we will take the series 2, 5, 8, 11, 14, writing it under itself in an inverse order, and add each term.
Here we perceive that 16, the sum of the extremes, multiplied by 5, the number of terms, equals 80, which is twice the sum of the series. Dividing 80 by 2 gives 40, which is the sum required.
Multiply the sum of the extremes by the number of terms, and divide the product by 2.
2. The extremes are 5 and 32, and the number of terms 12. What is the sum of all the terms? Ans. 222. 3. How many strokes does a common clock make in 12 hours? Ans. 78 strokes. 4. What debt can be discharged in a year by weekly payments in arithmetical progression, the first being $ 24, and the last $1224? Ans. $32448.
5. Suppose 100 apples were placed in a line 2 yards apart, and a basket 2 yards from the first apple. How far would a boy travel to gather them up singly, and return with each separately to the basket?
Ans. 20200 yards.
6. The first term of an arithmetical progression is 4, the common difference 5, and the number of terms 7. What is the sum of the series?
NOTE. First find the last term by 451, and then proceed as in 454.
7. The extremes of an arithmetical progression are 8 and 64, and the common difference is 8. What is the sum of the series? Ans. 288.
NOTE. First find the number of terms by 453, and then proceed as in 454.
455. By reversing some one of the four problems now given, or by combining two or more of them, all of the sixteen remaining problems of Arithmetical Progression may be solved or analyzed.
456. A Geometrical Progression is a series of numbers increasing or decreasing by a constant multiplier.
When the multiplier is greater than a unit, the series is ascending.
When the multiplier is less than a unit, the series is descending.
Thus, 2, 6, 18, 54, 162, is an ascending series, in which 3 is the multiplier. 162, 54, 18, 6, 2, is a descending series, in which the multiplier.
457. In every geometrical progression there are five parts to be considered, any three of which being given, the other two may be determined. They are as follows: The first term, last term, ratio, number of terms, and the sum of all the terms.
458. The first and last terms are the extremes, and the intermediate terms are the means.
459. To find any term, the first term, the ratio, and number of terms being given.
1. The first term of a geometrical ratio is 2 and the multiplier or ratio is 3. What is the 4th term?
33 = 27.
2 × 27 = 54, 4th term.
SOLUTION. The first term exists independently of the ratio. Since the number is multiplied by 3 the second term is 2 x 3, the third term 2×3×3, or 2 x 32, the fourth term 2 × 3 × 3 × 3 or 2 × 38. Using the ratio once as a factor, gives the second term; using it twice or its second power, the third term; using it three times or its third power, the fourth term. The third power of 3 is 27 and the first term 2 mul tiplied by 27 gives the 4th term 54.
· Multiply the first term by that power of the ratio denoted by the number of terms less 1.
2. The first term of a geometrical series is 4, the ratio is 3. What is the 9th term?
Ans. 4 x 38
3. The first term is 1024, the ratio, and the number of terms 8. What is the last term?
4. The first term of a geometrical progression is 6, the ratio 10, and the number of terms 5. last term?
What is the
5. A boy bought 9 oranges, agreeing to pay 1 mill for the first orange, 2 mills for the second, and so on. did the last orange cost him?
What Ans. $.256.
6. The first term is 7, the ratio, and the number of terms 7. What is the last term?
7. What is the amount of $1 at compound interest for 5 years, at 7% per annum? Ans. $1.40255+
In the above example the first term is $1, the ratio is $1.07, and the number of terms is 6.
8. A drover bought 7 oxen, agreeing to pay $3 for the first ox, $9 for the second, $27 for the third, and so What did the last ox cost him? Ans. $2187.
460. To find the sum of all the terms, the extremes and ratio being given.
1. Find the sum of a geometrical series in which the first term is 2, the last term 512, and the ratio 4.
8 +32 + 128 + 512 + 2048 = 2728 =
But 2+8+32 + 128 + 512 =
Four times the sum
of all the terms. Once the sum of all the terms. Three times the sum of all the terms, Once the sum of all the terms.
512 × 4 = 2048 2 2046 ÷ 3 = 682, sum.
SOLUTION. If we take the series 2, 8, 32, 128, 512, in which the ratio is 4, multiply each term by the ratio, and add the terms thus multiplied, we shall have the result shown in the operation.
The subtraction is performed by taking the lower line or series from the upper. All the terms cancel except 2048 and 2. Taking their difference, which is 3 times the sum, and dividing by 3, the ratio less 1, we must have the sum of all the terms.
RULE.-Multiply the greater extreme by the ratio, subtract the less extreme from the product, and divide the remainder by the ratio less 1.
Let every decreasing series be inverted, and the first term called the last; then the ratio will be greater than a unit. If the series is infinite, the first term is a cipher.
2. The first term is 2, the last term 486, and the ratio 3. What is the sum of all the terms?
Ans. 728. 3. The first term is 4, the last term 262144, and the ratio is 4. What is the sum of the series?
Ans. 349524. 4. The first term of a descending series is 162, the last term 2, and the ratio. What is the sum? Ans. 242. 5. What is the value of,,, etc., to infinity?
6. The extremes are 3 and 384, and the ratio is 2. What is the sum of the series? Ans. 765.
7. If the extremes are 5 and 1080, and the ratio is 6, what is the sum of the series?
8. If the first term is 44, the last term 485, and the ratio, what is the sum of the series? Ans. 777
461. Given the first term, the ratio, and the number of terms, to find the sum of the series.
1. The first term of a geometrical progression is 4, the ratio 3, and the number of terms 6. What is the sum of the series?