ARITHMETIC PROGRESSION Determining an Arithmetic Progression CHAPTER 1 : PROGRESSIONS 1) Determine whether each of the following sequences is an arithmetic progression. (a) 5, 9, 13, 17,... (b) -9, -7, -5, -3,... 1 1 (c) 20, 18, 17, 15, 14,... 2 2 (d) 2p, 3p, 5p, 8p,... 2) In the diagram, the areas of the right-angled triangles form an arithmetic progression. State its common difference. Term of an Arithmetic Progression 3) Find the eight term of the following arithmetic progressions. (a) 2, 6, 10, 14,... (b) -5, -2, 1, 4,... 4) a) Find the number of terms in the following arithmetic progression. -3, -8, -13,..., -163 b) Find the number of odd numbers between 10 and 100. 5) The first three terms of an AP are m, 2m-2 and 2m+1. Find (a) the value of m, (b) the 7 th term of the progression. 6) The twelveth and eighteenth terms of an AP are 50 and 74 respectively. Find (a) the first term and the common difference, (b) the 50 th and 100 th terms of the AP. 7) The volume of liquid in a container is 325 litres on the first day. Subsequently, 5 litres of liquid is added to the container every day. Calculate the volume, in litres, of liquid in the container at the end of the ninth day. The Sum of an Arithmetic Progression Sum of the first n terms of an arithmetic progression 8) The first term of an AP is 33 and the last of fourteenth terms is -15. Find the sum of these terms. Copyright www.epitomeofsuccess.com Page 1
9) Find the sum of the first eight terms of the AP -15, -9, -3,... 10) Find the sum of the first twelve terms of an AP if the first term is -7 and common difference is 3. 11) The sum of the first eight terms of an AP is 108 and the sum of the first sixteenth terms is 408. Find the first term and the common difference of the progression. 12) The fifth term of an AP is 24 and the sum of the first ten terms is 265. Find (a) the first term and common difference, (b) the sum of the first five terms. 13) The nth term of an AP is given by T n = 2n-3. Determine (a) the sum of the first twenty terms, (b) which term of the progression is equal to 27. Sum of a specific number of an AP 14) The first three terms of an AP are 12, 17 and 22. Find (a) the tenth term, (b) the sum of the next tenth terms. 15) The sum of the first eight terms of an AP is 100 and the sum of the next four terms is 122. Find (a) The first term and common difference, (b) The first five terms of the progression. The number of terms in AP 16) In an AP, the first term is 8 and the last term is 78. If the sum of these terms is 645, how many terms are there in this progression? 17) Given that the sum of the first n terms of an AP 8, 11, 14, is 294. Find the value of n. 18) The sum of the first n terms of an AP denoted by S n is given by (a) the value of the first term and the common difference, (b) the ninth term of the progression. S n 2n 2 3n. Find 2 19) The first three terms of an AP are k, 2k+1 and 5k-1. Find (a) the value of k, (b) the sum of the first seventh terms of the progression. 20) The first three terms of an AP are 7, 11 and 15. Find (a) the common difference of the progression, (b) the sum of the next seven terms. 21) The eight term of an AP is 4m+6 and the sum of the first five terms of the progression is 9m-4 where m is a constant. Given that the common difference of the progression is 4, find the value of m. Copyright www.epitomeofsuccess.com Page 2
Problems involving Arithmetic Progression 22) An arithmetic progression has eight terms. The sixth term is 15 and the sum of the odd terms is 44. Find (a) the first term and common difference, (b) the eight term. 23) The diagram shows a few sectors of concentric circles with centre O and angle rad. On 6 each subsequent sector, its radii are increased by 3 cm compared to the previous one. Given that the length of arc of the nth sector is 5 cm, find (a) the radius of the nth sector, (b) the value of n, (c) the sum of the radii of the first twelve sectors. 24) A piece of wire of length 40 cm is cut to form five circles as shown in the diagram. The diameter of each circle increases by 1 cm in sequence. Find (a) the diameter of the smallest circle, (b) the number of circle that can be formed if the length of the wire used is 105 cm 25) The diagram shows part of an arrangement of a flower vase of equal size. The number of vases in the bottom and top rows are 20 and 9 respectively. The height of each vase is 15cm. If each successive vase requires 1 vase less than the previous row, calculate (a) the height, in cm, of this arrangement, (b) the cost of the vases used if the cost of one vase is RM2.50. GEOMETRIC PROGRESSIONS Determining a Geometric Progression 26) State whether each of the following sequences is a geometric progression. (a) 1, 5, 25, 125,... 1 1 1 (b) 2,,,,... 2 4 8 (c) m, 3m 2, 6m 3, 9m 4,... Copyright www.epitomeofsuccess.com Page 3
27) The diagram shows a sequence of squares that are constructed. The first square PQRS has sides m cm in length. On each subsequent square its sides is half of the previous one. (a) Show that the areas of the squares form a GP. (b) State its common ratio. 16 28) Given a GP h, -4,, k,..., express k in terms of h. h 29) The first three terms of a sequence are 4, x, 16. Find the value(s) of x so that the sequence is (a) an arithmetic progression, (b) a geometric progression. Term of a Geometric Progression Specific terms in geometric progressions 30) a) Find the ninth terms of the GP 1, 3, 9, 27,... 1 1 1 b) Find the nth term of the GP 2,,,,... 2 8 32 31) Given that x-2, x-1 and 3x-5 are the first three terms of a GP, find (a) the possible values of x, (b) the eighth term of the GP for x>2. 32) The third and sixth terms of a GP are 36 and 972 respectively. Find (a) the first terms and common ratio, (b) the seventh term of a GP. 33) The second term of a geometric progression exceeds the first term by 4 and the sum of the second and third terms is 24. Find (b) the sixth term for each of the possible geometric progressions. Number of terms in geometric progressions 34) How many terms are there in the GP -2, 4, -8, 16,... 1024? 1 9 35) Which term of GP 18, 9, 4,... is? 2 32 36) Which is the first term of the GP 3, 6, 12,... to exceed 300? 1 37) Which is the first term of the GP 8, 4, 2,... that is less than? 100 Copyright www.epitomeofsuccess.com Page 4
The Sum of Geometric Progression To find the sum of the first n terms of a GP 38) Find the sum of the following geometric progressions. 1 1 1 (a),,,... up to the first six terms 8 4 2 (b) 1, -4, 16,..., -1024 39) If k-1, 2k-2 and 3k-1 are the first three terms of a GP where k>1, find (a) the value of k, (b) the sum of the first nine terms of the GP. 40) The third and sixth terms of a GP are 108 and -32 respectively. Find (b) the sum of the first four terms. 41) The nth term of a GP is given by T n =2 7-n. Find (b) the sum of the first five terms. To find the sum of a specific number of terms in GP 42) Find the sum of the GP -2, 6, -18,... from the fifth to the 10 th terms, inclusive. 43) The first three terms of the GP are 1, 3, 9,... Find the sum of the next five terms. 44) The sum of n terms of a GP is given by S n = 1-2 n. Find (a) the sum of the first seven terms, (b) the sum of the next three terms. To find the number of terms in a GP 45) The sum of the first n terms of a GP whose first term is 2 and the common ratio is 4 is 682. Find the value of n. 46) Find the least number of terms of the GP 2, 6, 18,... which must be taken for the sum to exceed 4000. 47) The sum of the first n terms of the GP 6, 12, 24,... is 1530. Find (a) the common ratio of the progression, (b) the value of n. Sum to Infinity of a Geometric Progression 48) Find the sum to infinity of the following geometric progressions. Copyright www.epitomeofsuccess.com Page 5
4 8 (a) 18, 6, 2,... (b) 2,,,... 3 9 49) The sum to infinity of a certain GP is 36. If the first term is 12, find the common ratio. 50) The sum to infinity of a GP with common ratio 3 1 is 27. Find the first term. 51) The third and second terms of a GP are (b) the sum to infinity of this GP. 2 8 2 and respectively. Find 3 81 52) Represent each of the following recurring decimals as the fraction in its simplest form. (a) 0.454545... (b) 2.6 1 2 53) In a GP, the common ratio is and the third term is 2. Calculate 3 3 (a) the first term, (b) the sum to infinity of the GP. 54) Express the recurring decimal 5.363636... as a mixed fraction. 55) The second term of a GP is 30. The sum of the second and third terms is 55. Find (a) the first term and the common ratio of the GP, (b) the sum to infinity of the GP. Problems involving Geometric Progressions 56) A ball is dropped from a height of 6m above a flat surface as shown in the diagram. Each time the ball hits the surface, it bounces up to 3 2 of the previous height. Find (a) The height that the ball bounces up after it hits the surface for the tenth time, (b) The total distance the ball travels before it stops bouncing. Copyright www.epitomeofsuccess.com Page 6
57) The diagram shows a few semicircles that touch each other at P. S is the midpoint of PR, T is the midpoint of PS, U is the midpoint of PT and so on. (a) Show that the areas of the semicircles with radii PS, PT, PU... form a GP and state its common ratio. (b) If PR = 18cm and this sequence of the semicircles is constructed indefinitely, find the sum of the areas of all semicircles. 58) The diagram shows the arrangement of the first three of an infinite series of similar triangles. The base and height of the first triangle are x cm and y cm respectively. The measurements of the base and height of each subsequent triangle are half of the measurements of its previous one. (a) Show that the areas of the triangle form a GP and state its common ratio. (b) Given that x=8cm and y=12cm, 3 2 (i) determine which triangle has an area of cm, 4 (ii) find the sum to infinity of the areas, in cm 2, of the triangles. Copyright www.epitomeofsuccess.com Page 7