Progressions Knowledge Examples

 Lets assume that the student asked what the solution to the below questions.   Questions: 1.    Is the sequence an arithmetic progression (AP)? 2.    Is the sequence a geometric progression (GP)?            3.    Show that the sequence  is an arithmetic progression and find the nth term.   4.    Find the sum of 20 terms of the arithmetic progression 1, 2, 3, 4, … 5.    Show that the sequence is a geometric progression and find the nth term. 6.    Find the sum of 8 terms of the geometric progression     7.    Find the geometric mean of 3 and 27, and show that the resultant sequences are both geometric sequences.   You can solve these questions by seeing the solutions to the given questions which also use the same concepts.     Questions which will help you to solve your questions   Question 1: Determine whether the following sequences are arithmetic progressions?       a) 5, 7, 10, 14, …                  b) 10, 6, 2, -2,  …                  c) Answer: If a sequence is an arithmetic progression (AP), then the difference between consecutive terms must be same. In other words the difference between the terms 2 and 1 must be same as the difference between the terms 4 and 3. a) 5, 7, 10, 14, …         Hence, the sequence is not an AP. b) 10, 6, 2, -2,  …   Hence the common difference is -4, and the sequence is an AP. c)     The common difference is ,              hence the sequence is an AP.   Question 2: Determine whether r the following sequences are arithmetic progressions?      a) …                     b) 2, 4, 12, 48, …                   c) 8, -4, 2, -1, … Answer: If a sequence is an geometric progression (GP), then the ratio between consecutive terms must be same. In other words a) …     The common ratio is  and the sequence is a GP. b) 2, 4, 12, 48, …     There is no common ration hence the sequence is not a GP. c)   8, -4, 2, -1, …      Hence the common ratio is  and the sequence is a GP. Question 3: Show that the sequence  is an arithmetic sequence and find the nth term.           Answer: First of all, find the differences between consecutive terms.  and  The differences between consecutive terms equal to each other and the common differences is . The sequence is an AP. The nth term is  where a is the first term and d is the common difference. Hence the nth term is     What is the common difference in your arithmetic series and what is the first term?   Question 4: Find the sum of 20 terms of the arithmetic progression 10, 12, 14, 16, … Answer: The equation for the sum of n terms of an AP is , where a is the first term of the progression and d is the common difference. The first term of the sequence is 10 and the common difference is 2. Hence the sum of 20 terms is . Question 5: Show that the sequence is a geometric progression and find the nth term.   Answer: For the sequence to be a GP the ratio between the consecutive terms must be same.          and                Hence, the sequence is a GP. The equation for the nth term for a GP is  where a is the first term of the progression and r is the common ratio. The first term of the progression is 2m and the common ratio is . Substituting these values in the equation of the nth term gives . Question 6: Find the sum of 11 terms of the geometric progression Answer: The equation for the sum of n terms of a GP is  where a is the first term and r is the common ratio. The common ratio is the ratio of two consecutive terms.  and the first term is 2. Hence the sum of 11 terms is .   What are the common ratio and the first term in your geometric series?   Question 7: Find the geometric mean of 16 and 4, and show that the resulting sequences are both geometric sequences Answer: The geometric mean, b, of two numbers a and c is . The two numbers are 16 and 5 and their geometric mean is                    The first progression is 16, 8, 4, … and the second progression is 16, -8, 4, … The common ratio for the first progression is  and for the second is .