Sequences, their sums and Induction Example (1) Calculate the 15th term of arithmetic progression, whose initial term is 2 and common differnce is 5. And its n-th term? Find the sum of this sequence from the initial term to the n-th term. (2) Let {a n } be an arithmetic progression whose 10th term is 67 and the 20th term is 137. Find the initial term and the common difference of this sequence. Then find its 100th term. Find the sum of this sequence from the initial ten to the 100th term. [1] Given following arithmetic progressions, whose initial term is a and its common difference is d. Find the n-th term and the sum from the initial term to the n-th term. (1) a = 100, d = 5 (2) a = 2, d = 1 (3) a = 23, d = 3 (4) a = 1, d = 1 [2] Let an arithmetic progression, whose common difference is 4, and the sum of its initial term and the 45-th term is 130. Find the sum from its initial term to 50th term. [3] Find the sum of all rational numbers between 100 and 200, which are divisible by 3. And find the sum of all rational numbers between 100 and 200, which are divisible by 3 but non-divisible by 5. [4] Let {a n } be an arithmetic progression, where a 19 = 230, a 25 = 220. Find n, when the sum from its initial term to the n-th term becomes maximum and find its sum. 1
Example 2 Calculate the 10th term and the n-th term of geometric progression, whose initial term is 3 and common ratio is 2. And find the sum from its initial term to the n-th term. [5] Find the n-th term of following geometric progressions, whose initial term is a ann its common ratio is r. And find the sum from its initial term to the n-th term. (1) a = 2, r = 3 (2) a = 1 4, r = 1 (3) a = 3, r = 1 2 (4) a = 4, r = 1 [6] Find the n-th term of the geometric progression, whose 5th term is 3 16 3 128. and its 8th term is [7] Given a geometric progression, whose sum from its initial term to the n-th term is 24 and the sum from its initial term to the 2n-th term is 30. Find its sum from the initial term to the 3n-th term [8] Find a, b, c, where the sequence a, b, c is an arithmetic progression, the sequence b, c, a is a geometric progression and its product abc = 125. 2
Example 3 [1] Find the sum of following sequences from its initial term to the n-th term. (1) 1 2, 4 2, 7 2, 10 2, (2) 1 4, 2 5, 3 6, 1 (3) k(k + 1) (4) kx k [2] Find the n-th term of the following sequence. 2, 7, 16, 29, 46, 67, [9] Find the sum of following sequences from its initial term to the n-th term. (1) 1 2, 3 2, 5 2, 7 2, (2) 1, 1 + 2, 1 + 2 + 3, (3) 1 2 3, 3 4 5, 5 6 7, [10] Find the following sums. (1) (k + 2 k ) 2 (2) (3) (k + 1)3 k 1 (4) 1 (k + 1)(k + 3) 1 k(k + 1)(k + 2) [11] Find the n-th term of the following sequence. (1) 4, 12, 26, 46, 72, (2) 1, 2, 5, 12, 27, 3
Example 4 [1] Given the sequence 1 2, 2 3, 1 3, 3 4, 2 4, 1 4, 4 5, 3 5, 2 5, 1 5, 5 6,. (1) Find n, where 18 is the n-th term. 25 (2) Find the 666th term. (3) Find the sum of this sequence from the initial term to the 666th term. [2] Let S n = 2 n n the sum of the sequence {a n } from its initial term to the n-th term. Find the n-th term of {a n }. [12] Grouping the sequence of odd numbers as 1 3, 5 7, 9, 11 13, 15, 17, 19 21,. (1) Find the 5th number of the 13th group. (2) Find the n-th number of the m-th group. where m m (3) Find m, n, as 2005 is the n-th number of the m-th group. [13] Given the sequence as 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5,, 1, 2,, k, 1, 2,, k, k + 1, (1) Find the term which is the first 10. (2) Find the 100th term. (3) Find the term of the n-th times 1. [14] Let S n be the sum of sequence {a n } from its initial term to the n-th term and S n = n 2 + 4n Find the n-th term of {a n }. 4
Example 5 Find the n-th term of following sequences. (1) a 1 = 3, a n+1 = a n 2 (n = 1, 2, 3, ) (2) a 1 = 3, a n+1 = 2a n (n = 1, 2, 3, ) (3) a 1 = 3, a n+1 = a n + 2n (n = 1, 2, 3, ) (4) a 1 = 3, a n+1 = 2a n 5 (n = 1, 2, 3, ) (5) a 1 = 3, a n+1 = 2a n + 3 n (n = 1, 2, 3, ) (6) a 1 = 3, a n+1 = 2a n 3n (n = 1, 2, 3, ) [15] Find the n-th term of following sequences. (1) a 1 = 1, a n+1 = a n + 5 (n = 1, 2, 3, ) (2) a 1 = 2, a n+1 = 2a n (n = 1, 2, 3, ) (3) a 1 = 1, a n+1 = a n + 2n 1 (n = 1, 2, 3, ) (4) a 1 = 1, a n+1 = 5a n 2 (n = 1, 2, 3, ) (5) a 1 = 1, a n+1 = 2a n + 3 (n = 1, 2, 3, ) (6) a 1 = 1, a n+1 = 3a n 2n + 5 (n = 1, 2, 3, ) [16] Let {a n } be a sequence, where a 1 = 1 2, a n+1 = 2a n a n + 1 (n = 1, 2, 3, ). (1) Let b n = 1 a n. Find b n+1 as a function of b n. (2) Find b n as a function of n. (3) Find a n as a function of n. [17] Let {a n } be a sequence, where a 1 = 1, a 2 = 2, a n+2 = 4a n+1 3a n (n = 1, 2, 3, ). (1) Find α β, where a n+2 αa n+1 = β(a n+1 αa n. (2) Find a n as a function of n. 5
Example 6 Prove by induction (cos θ + i sin θ) n = cos nθ + i sin nθ on positive integer n. [18] Prove the following inequality by induction, where n is a positive integer. 1 + 1 2 + 1 3 + + 1 n < 2 n [19] Let a n be a sequence defined as a 1 = 0, a n+1 = 1 + a n 3 = a n (n = 1, 2, 3, ). (1) Find a 1, a 2, a 3. (2) Find a n from the result of (1). (3) Prove (2) by induction. 6
Exercises [1] Find the three numbers, which make an arithmetic progression, its sum is 27 and its product is 704. [2] Let {a n } be an arithmetic progression whose initial term is a 1 and its common difference is d. Let {b n } be a geometric progression whose initial term is b 1 and its common ratio is r. b 1 \= 0, r > 0. Define the sequence {c n } as c n = a n b n (n = 1, 2, 3, ) and c 1 = 8, c 2 = 1, c 3 = 1. (1) Find r. (2) Find the n-th term of {c n }. And find the sum of {c n } from its initial term to the n-th term. [3] Let {a n } be an arithmetic progression whose initial term is 1 and its common difference is 4, and let {b n } be an airtimetic progression whose initial term is 2 and its common difference is 2. Let {c n } be the sequence ordering every terms of {a n } and {b n } from the sammest to the largest. Find c 100 and find c 1 + c 2 + c 3 + + c 100. [4] How many integers divisiblee by 36 from 1 to 710? How many integers divisible by 12 but non-divisible by 18 from 1 to 710? And their sum? 7
[5] Let {a n } be an arithmetic progression whose initial term is 1 and its common difference is 4 3. (1) Find the n-th term of {a n }. Hence find the sums a k and (2) Let {b n } be b n = [a n ]. Find n when b n = 103. And find the sum of all b n which satisfy b n 50. Find 3m b k when m is positive integer. n a k 2. [6] Make a sequence ordering {1}, {2, 3}, {3, 4, 5}, {4, 5, 6, 7}, {5, 6, 7, 8, 9}, as 1, 2, 3, 3, 4, 5, 4, 5, 6, 7, 5, 6, 7, 8, 9, (1) Find n, where the n-th term is the first 99. (2) The 1999th term is the l-th number of the k-th group. Find k and l, where let {1} be the first group {2, 3} be the second group {3, 4, 5} be the third group. [7] Let S n be the sum of an arithmetic progression {a n } (n = 1, 2, ) from its initial term to the n-th term. If {S 1, S 2, S 3 } = {22, 21, 20}, find the n-th term of a n. [8] Let S n be the sum of an arithmetic progression whose initial term is 1 and its common difference is a positive integer d. Let n 3 (1) Find n d, as S n = 94 (2) Explain why there do not exist n and d as S n = 98. 8
[9] Calculate S n = n 1 k(k + 1)(k + 2)(k + 3). [10] Let S n be the sum of sequence {a n } from its initial term to the n-th term. And let S 1 = 0, S n+1 3S n = n 2 (n = 1, 2, 3, ). (1) Find the relation between a n and a n+1. (2) Find the n-th term of {a n }. [11] Let S n be the sum of sequence {a n } from its initial term to the n-th term. And let S n = n 3 + 21n 2 + 65n (n = 1, 2, 3, ). (1) Find the initial term a 1 of {a n }. (2) Find the n-th term a 1 of {a n }. (3) Find n where a n > 151. [12] Let the probability of winning of a game be 1 2. Starting this game with x coins and if you win you receive one coin and if not you lose one coin. If you get 10 coins or no coins, the games are over. Let p(x) (0 < x < 10) be the probability which you have x coins and let p(0) = 1, p(10) = 0. (1) Write down p(x) by using p(x + 1) and p(x 1). (2) Find p(1). 9