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From the conditions of the question, we know that at the end of the first year there will be simply one stalk or branch, which we will denote by 1,; at the end of the second year, this branch will become one year old, and produce a new branch, so that we shall have 1, +1,; at the end of the third year the branches 1, +1, will become 1, +17, the first of which being 2 years of age, will produce two new branches, the other will produce one new one; we shall therefore have 1, +1, +3,. Proceeding in this way, we obtain the follow. ing results :
End of 1st year we have 1. 6 2
6 1,+1. 66 36
1, +1, +3,
1, +1, +3, +8. -66 5 6 6 1. +1, +3, +8, +21. =34
5 6 " " 15+14+3,+8, +21, +55,=89
In this scheme, the small figures at the bottom of the larger ones, denote the age in years, of the branches to which they are attached. Thus, at the end of the fifth year, there will be one branch 4 years old, one branch 3 years old, three branche es 2 years old, eight branches 1 year old, and twenty-one new branches of no age.
The law of the above series is obvious. It is such, that twice any term, increased by the sum of all the preceding terms, gives the next succeeding term.
These terms may be found most easily by continual addi. tion, as given on the following page, where each succeeding term is found by adding the two preceding ones.
now branches 6th year,
new branches 7th year,
new branches eth year,
new branches 9th year,
new branches 10th yoar,
new branches 11th year,
now branches 12th year,
now branches 13th year,
total branches 6th year,
144 total branches 7th year,
377 total branches 8th year,
987 total branches 9th year, 1597
2584 total branches 10th year, *4181
6765 total branches 11th year, 10946
17711 total branches 12th year, 28657
46368 total branches 13th year, 75025
121393 total branches 14th year,. 196418
317811 total branches 15th year, 514229
832040 total branches 16th year, 1346269
2178309 total branches 17th year, 3524578
5702887 total branches 18th year, 9227465
14930352 total branches 19th year, 24157817
39088169 total branches 20th year. 63245986
new branches 14th year,
new branches 15th year,
now branches 16th year,
Bew branches 17th year,
now branches 18th year,
new branches 19th year,
new branches 20th year,
MISCELLANEOUS QUESTIONS. 93. What are the prime factors of 2006 ? 2. What are the prime factors of 3742?
3. What is the greatest common measure of 720, 360, and 180 ?
4. What is the greatest common measure of 420, 147, and 210?
5. What is the least common multiple of 4, 16, 24, and 40 ? 6. What is the least common multiple of 8, 36, and 100 ? 7. What are all the divisors of 376 ? 8. What are all the divisors of 23456 ? 9. What is the sum of the divisors of 7866 ? 10. What is the sum of the divisors of 1000 ? 11. Reduces to its lowest terms. 12. Reduce 338461 to its lowest terms. 13. Reduce the improper fraction 1936 to a mixed fraction. 14. Reduce 45667 to a mixed fraction. 15. Reduce 67 jí to an improper fraction. 16. Reduce 377 to an improper fraction. 17. What is the product of into 14? 18. What is the product of j. into *?
19. Reduce the compound fraction of g of 4 of 14 to a simple fraction.
20. Reduce it, it, and 4 to fractions having a common denominator.
21. What is the sum of 1'5, 4, and ? 22. What is the quotient of divided by f? 23. Reduce the complex fraction to its simplest form. 24. What is the value of 24 of a mile? .
25. Can the vulgar fraction i be accurately expressed in decimals?
26. How many places of decimals will be required to express ?
27. Find the compound repetend equivalent to sto
31. Find some of the approximative values of the continued fraction
3+ &c. 32. What must be the length of a thread, which will wind spirally about a cylinder of 4 feet in circumference, and 60 feet in length, the distance between each turn of the thread, being 1 foot ?
33. Required to divide the number 90 into four parts ; such that, if the first be increased by 5, the second decreased by 4 the third multiplied by 3, and the fourth divided by 2, the re. sults, in each case, shall be the same.
34. If A can perform a piece of work in 10 days, B in 12 days, C in 16 days, then how many days will be required for all together to perform the work?
35. A shepherd, in the time of war, fell in with a party of soldiers who plundered him of half his flock, and half a sheep over; afterwards a second party met him, who took half of what he had left, and half a sheep over; and soon after this, a third party met him, and used him in the same manner; and then he had only five sheep left. It is required to find what number of sheep he had ai first.
36. Four persons, A, B, C, and D, spent 20 shillings in com. pany ; when A proposed to pay }, B , C }, and D i part; but when the money came to be collected, they found it was not sufficient to answer the intended purpose. The question then is, to find how much each person must contribute to make up the whole reckoning, supposing their several shares to be to each other in the proportion above specified.
37. If of a pound of cinnamon is worth 18 cents, what will 7 pounds cost ?
38. If a family of 8 persons spend $480 in 32 months, how much would 16 persons spend in 8 months?
39. Two persons depart from the same place at the same time; the one travels 30, the other 35 miles a day. How far distant are they after 7 days, if they travel both in the same direction ; and how far if they travel in contrary directions ?
40. A stationer sold quills at 10s. 6d. a thousand, by which he cleared ß of his money; but growing scarce, he raised them to 128. a thousand. What did he clear per cent. by the latter price?
41. How much can a person give for an annuity of $400, which has to run 12 years, if the interest be reckoned at 3 per cent. ?