SOLUTION: If 1,1,3,9 be added respectively to four terms of an AP.,a GP results. Find the four terms of the AP
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Question 1208641: If 1,1,3,9 be added respectively to four terms of an AP.,a GP results. Find the four terms of the AP
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!
Answer: 1, 3, 5, 7
Explanation
AP = arithmetic progression
GP = geometric progression
We have these 3 sequences
| Sequence 1 | 1 | 1 | 3 | 9 |
| Sequence 2 | a | a+d | a+2d | a+3d |
| Sequence 3 | 1+a | 1+a+d | 3+a+2d | 9+a+3d |
Sequence 3 is the sum of sequence 1 and sequence 2.
Add straight down.
Because we're told that sequence 3 is geometric, dividing any term over its previous term will get us the common ratio r.
r = (2nd term)/(1st term) = (1+a+d)/(1+a)
r = (3rd term)/(2nd term) = (3+a+2d)/(1+a+d)
Equate those expressions to form this equation
(1+a+d)/(1+a) = (3+a+2d)/(1+a+d)
Solving for 'a' leads to a = 0.5d^2 - 1
I'll let the student do the scratch work.
Furthermore,
r = (3rd term)/(2nd term) = (3+a+2d)/(1+a+d)
r = (4th term)/(3rd term) = (9+a+3d)/(3+a+2d)
Equating them gives us (3+a+2d)/(1+a+d) = (9+a+3d)/(3+a+2d)
Plug in a = 0.5d^2 - 1 and we get
(3+0.5d^2 - 1+2d)/(1+0.5d^2 - 1+d) = (9+0.5d^2 - 1+3d)/(3+0.5d^2 - 1+2d)
Solving that equation yields d = 2
Plug d = 2 into a = 0.5d^2 - 1 to get a = 1.
To summarize:
a = 1
d = 2
This table
| Sequence 1 | 1 | 1 | 3 | 9 |
| Sequence 2 | a | a+d | a+2d | a+3d |
| Sequence 3 | 1+a | 1+a+d | 3+a+2d | 9+a+3d |
Updates to
| Sequence 1 | 1 | 1 | 3 | 9 |
| Sequence 2 | 1 | 3 | 5 | 7 |
| Sequence 3 | 2 | 4 | 8 | 16 |
Sequence 2 is arithmetic because we add 2 to each term to get the next term.
The nth term of this sequence is 2n-1 where n is an integer that starts at n = 1.
Sequence 3 is geometric since dividing any given term over its previous term results in the same common ratio
4/2 = 2
8/4 = 2
16/8 = 2
Put another way: we double each term to get the next term.
The nth term of this geometric sequence is 2^n.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!
Let the four terms of the AP be a, a+d, a+2d, and a+3d.
Then the four terms of the GP are a+1, a+d+1, a+2d+3, and a+3d+9.
In a GP, the ratio of consecutive terms is a constant.
first, second, and third terms...
[1]
second, third, and fourth terms...
[2]
Solve [1] and [2] simultaneously
The AP has first term 1 and common difference 2.
ANSWER: 1, 3, 5, 7
CHECK: The GP is 1+1=2, 3+1=4, 5+3=8, and 7+9=16
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