Answer:
Step-by-step explanation:
In arithmetic progression common difference is given by
d = nth term - (n-1)th term.
In geometric progression common ratio is given by
r = nth term / (n-1)th term.
Another thing important to note is that in both AP and GP common difference and common ratio remains same for any two consecutive terms which can be calculated using above formula.
we will be using this concept to solve this problem.
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Let look at the problem
series
11, 6 ,1
Lets try to find common difference for this series
common difference of third and second term = 1-6 = -5
common difference of second term and first term = 6 -11 = -5
Thus, we see that common difference is same in both case ,
hence it can be concluded that series is Arithmetic progression.
another way to validate this is
nth term of AP is given by
nth term = a + (n-1)d
where a is first term
lets try to find third term using this formula and using d = -5 and a = 11
3rd term = 11 + (3-1)-5 = 11 + 2*-5 = 11- 10 = 1
Thus, we see that third term calculated is same as given in series i.e 1
Hence series is Arithmetic progression validated as well.
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Problem is solved but lets see if the series is GP or not
common ratio of third and second term = 1/6 = 1/6
common ratio of second term and first term = 6 /11
Thus, we see common ratio is not same for the two consecutive term in the series and hence series cannot be geometric progression.
