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.
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, …
a) … The common ratio is and the sequence is 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.
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 .
