a + 2d, a + 3d, where the coefficient of d is always 1 less than the number of the term. If the number of terms is n and the last term l, 1 = a + (n − 1)d EXERCISE 126 1. Find the 9th term of the series 5, 8, 11, 14. Consider the 9th term, the last term of the series. The common difference is 8 The first term a, is 5. 5 = 3. Substitute these values in the formula, l = a + (n − 1)d. 2. Find the 10th term of the series, 3, 5, 7, 9, 11, 3. Find the 15th term of the series, 4, 7, 10, 13, 4. Find the 13th term of the series, 10, 8, 6, 4, 5. Find the 20th term of the series, 1, 1, 12, 13, 6. Find the 10th term of the series, 50, 46, 42, 38, 7. Find the 12th term of the series, x, 4x, 7x, 10x, 8. Find the 8th term of the series, a- 4b, a 2b, a, a + 26, 9. Find the 7th term of the series x + 4ÿ, x + 3y, x+2y, 10. A body, falling from rest, falls 16 feet the first second, 3 times as far during the second second, 5 times as far during the third second and so on. How far does it fall during the 6th second? SUM OF THE SERIES 174. If a represents the first term of an A. P., d the common difference, I the last term, n the number of terms and s the sum of the n terms, s = a + (a + d) + (a + 2d) + (a + 3d) + + (ld) + 1 ors ( − d) + (l − 2d) + (l3d) + = l + . +(a + d) + a (Writing the terms in the reverse order) Add these two equations, (a + 1) + (a + 1) + (a + 1) + (a + 1) + + (a + 1) + (a +1) Since the sum of any two corresponding terms is a + l and there are n terms in each series, involve the five numbers, a, d, l, n, s. If any three of these are known, the other two may be found from these equations. EXERCISE 127 1. Find the sum of 15 terms of the series, 2, 21, 21, . . . As neither of the two formulæ contains the unknown and the three known numbers, we must first find l. 9. A body falls 16 feet the first second, 3 times as far the second second, 5 times as far the third second and so on. How far does it fall in 6 seconds? 10. An automobile coasting down a hill, reaches the foot of the hill in 8 seconds. If it goes 10 feet the first second, 15 feet the second second, 20 feet the third second and so on, how long is the hill? Since n is neither given nor required, eliminate n from these two simultaneous equations. 14. How many terms in the series, 3, 5, 7,....71? 222? .if the sum is 17. Find the extremes of the series...27, 32, 37,..... if the sum of 17 terms is 714. 18. Find an A. P. of 10 terms, whose sum is -15, whose second term is 9 and whose sixth term is -3. Derive the formula for: 19. l, given a, n, s. 20. s, given d, n, l. 21. a, given d, n, s. 22. d, given a, l, s. 23. n, given a, d, s. 24. Insert 4 arithmetical means between 3 and 6. Given a = 3 since = a + (n x ι = 6 Hence the series is, 3, 33, 41, 4, 5, 6. 25. Insert 7 arithmetical means between 1 and 5. 26. Insert 10 arithmetical means between 20 and 3. 27. Insert 8 arithmetical means between 5 and −5. 28. Insert 5 arithmetical means between x + 2y and When three numbers are in A. P., the second term is the arithmetical mean between the first and third. In solving problems in A. P. involving two unknowns, the series may be represented by x, x + y, x + 2y, etc. When the number of terms is small, a symmetrical form is more convenient. In each case, the sum of the terms involves only one unknown. 33. The sum of three numbers in A. P. is 27 and their product is 585. Find the numbers. 34. The sum of three numbers in A. P. is 30 and the sum of their squares is 308. Find the numbers. |
