Question 1164214: When the terms of a Geometric Progression(G.P) with common ratio r=2 is added to the corresponding terms of an Arithmetic (A.P), a new sequence is formed. If the first terms of the G.P and A.P are the same and the first three terms of the new sequence are 3, 7, and 11 respectively, find the nth term of the new sequence.
Found 2 solutions by greenestamps, jim_thompson5910:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
The statement of the problem is faulty....
We are told that the first terms of the AP and GP are the same; and we are told that the sum of those two terms is 3. So the first term of each sequence is 3/2.
We are told the common ratio for the GP is 2; so the first three terms of the GP are 3/2, 3, and 6.
We are told the first three terms of the new sequence are 3, 7, and 11.
That means the first three terms of the AP are 3-3/2 = 3/2, 7-3 = 4, and 11-6 = 5.
But 3/2, 4, 5 is not an AP.
Answer by jim_thompson5910(35256)  (Show Source):
You can put this solution on YOUR website!
The tutor @greenestamps has a great way to see why your teacher has most likely made a typo/mistake of some kind. Here's another clue that the problem is faulty.
"the first three terms of the new sequence are 3, 7, and 11" means the new sequence is arithmetic. Each time we're adding 4 to get the next term. This is of course assuming the pattern holds up.
But adding a GP and AP together does not lead to an arithmetic sequence unless the common ratio of the GP is r = 0 or r = 1.
Consider the sequence {0,0,0,0,...} which is geometric with r = 0. Adding this to any AP leads to the same AP. This isn't too exciting, but it's still important to know. If r = 1, then something like {4,4,4,...} is geometric which means we shift all of the terms of an AP by 4, meaning the new sequence is also an AP as well.
But for r = 2 and larger is when the sum is no longer arithmetic.
Something like {1,2,4,8,...} will add onto an AP to make some non-arithmetic sequence. This is because the gap from 1 to 2 is smaller than the gap from 2 to 4, and the gap widens each time. There's no way to have a fixed width the entire time. This will reflect in the new sequence as well.
I would ask your teacher for clarification.
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