Sequences #1 - Questions and Answers

Question 1 Read the numbers and decide what the next number should be. 5 15 6 18 7 21 8 Answer 15 = 5 * 3 the second term is the first term multiplied by 3. 18 = 6 * 3 the fourth term is the third term multiplied by 3. 21 = 7 * 3 the sixth term is the fifth term multiplied by 3. Following this pattern, the eighth term in the series is predicted to be the seventh term times three. It is stated that the sequence's seventh term will equal 8. 8*3=24 The next number to be 24. Question 2 What is the difference between arithmetic and geometric sequences? Answer 1 Arithmetic sequence is a sequence in which term differs by a constant number. This constant number is called as common difference. For example: 2, 4, 6... is an arithmetic sequence. a2-a1 = 4-2 = 2 a3 - a2=6-4 = 2 Difference between two successive terms is equal. So, it is an arithmetic sequence in which common difference is 2. Geometric sequence is sequence in which successive term is multiple of common ratio. Sequence a1, a2, a3, ... is geometric sequence if: a2/a1 = a3/a2 Common ratio is r = a2/a1 So, the difference between an arithmetic sequence and geometric sequence is that an arithmetic sequence has a constant difference between each term and a geometric sequence has a constant ration between each term. Answer 2 Step 1. Arithmetic Sequence: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference (d). The general form of an arithmetic sequence is given by the formula: an = a1 + (n-1)d where an represents the nth term, a1 is the first term, n is the position of the term, and d is the common difference. Step 2. Geometric Sequence: A geometric sequence is a sequence of numbers in which each term is obtained by multiplying the preceding term by a constant ratio. This constant ratio is called the common ratio (r). The general form of a geometric sequence is given by the formula: an = a1r^(n-1) where an represents the nth term, a1 is the first term, n is the position of the term, and r is the common ratio. In summary, arithmetic sequences have a constant difference between terms, while geometric sequences have a constant ratio between terms. Answer 3 The difference between arithmetic and geometric sequences can be summarized as follows: Arithmetic Sequence: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is always the same. This common difference is denoted by d. In an arithmetic sequence, each term can be found by adding the common difference to the previous term. The general form of an arithmetic sequence is given by: a1, a1 + d, a1 + 2d, a1 + 3d, ... where a1 represents the first term of the sequence. Geometric Sequence: A geometric sequence is a sequence of numbers in which each term is obtained by multiplying the previous term by a constant ratio. This common ratio is denoted by r. In a geometric sequence, the ratio of any term to its previous term remains constant. The general form of a geometric sequence is given by: a1, a1Xr, a1Xr², a1 Xr³, ... where a1 represents the first term of the sequence. To summarize the differences: Arithmetic sequences have a constant difference between consecutive terms (d), whereas geometric sequences have a constant ratio between consecutive terms (r). In an arithmetic sequence, each term is obtained by adding the common difference to the previous term, while in a geometric sequence, each term is obtained by multiplying the previous term by the common ratio. The general form of an arithmetic sequence is a1, a1 + d, a1 + 2d, a1 + 3d,..., whereas the general form of a geometric sequence is a1, a1 Xr, a1Xr², a1 Xr3,.... Remember that a1 represents the first term of the sequence, d represents the common difference in an arithmetic sequence, and r represents the common ratio in a geometric sequence. Question 3 Write the recursive formula for each geometric sequence. 640, 160, 40, 10, 2.5, ... Answer Use the recursive formula of a geometric sequence given by: an = r(an - 1) where r is the common ratio. Find the common ratio: r = 160/640 = 40/160 = 10/40 = 2.5/10 = 1/4 Substibute r = 1/4: an = 1/4(an-1) the recursive formula is an = 1/4(an - 1), where the initial condition is a1 = 640 Question 4 Write a recursive rule and an explicit rule for an arithmetic sequence that models a situation. Then use the rule to answer the question. Jack begins an exercise routine for 10 minutes each day. Each week he plans to add 10 minutes per day to his exercise routine. For how many minutes will he exercise for each day on the 7th week? Answer If Jack exercises for 10 minutes for the first day, then the first term of the sequence is a1 = 10. If he adds 10 minutes to his exercise routine each day, then the sequence is arithmetic since it has a common difference of d=10. The recursive formula for an arithmetic sequence is an = an-1+d, n ≥ 2. Since d = 10, the the recursive formula is then a1 = 10, an = an-1 +10, n ≥ 2 The explicit formula for an arithmetic sequence is an = a1 + (n - 1)d. Since a1 = 10 and d=10, then an = 10 + (n - 1)(10). Disturbing the 10 gives an = 10 + 10n-10. Combining like terms then gives an = 10n. The 7th week is when n = 7. Substituting n = 7 into an = 10n the gives a7 = 10 × 7 = 70. He will then exercise for 70 min in the 7th week. Question 5 Find a recurrence relation for the number of bit sequences of length n with an even number of Os. Answer Let an represent the number of bit sequences of length n with an even number of Os.