the first, second, and fifth term of an AP are three consecutive terms of an exponential sequence. if the first term of the linear sequence is 7 ,find it common difference
How did they define "exponential sequence" in your course ?
Assuming you mean Geometric Sequence, we have
For The AP, a=7, so we want d.
The ratio between terms of a GP is constant, so we form the ratio between successive given terms of the AP:
(7+d)/7 = (7+4d)/(7+d)
d=14 or d=0
Now, d=0 is not very interesting:
AP = 7,7,7,7,7...
GP = ... 0,0,0, ...
so it doesn't work anyway.
With d=14, we have the AP
7,21,35,49,63
And the named terms form the terms of the GP
... 7, 21,63, ...
with r=3
Let's denote the terms of the arithmetic progression (AP) as follows:
First term = a
Common difference = d
We know that the first, second, and fifth terms of the AP are three consecutive terms of an exponential sequence. This means that they follow the pattern: a, a*r, and a*r^4, where r denotes the common ratio.
We can set up the following equations based on this information:
a + d = a*r ...(1)
a + 2d = a*r^4 ...(2)
From equation (1), we can isolate a in terms of d:
a = a*r - d
Substituting this value of a in equation (2), we get:
(a*r - d) + 2d = (a*r)^4
ar - d + 2d = (ar)^4
-ar + 3d = (ar)^4
Since we are given that the first term of the linear sequence is 7, we can substitute a = 7 into the equation above:
-7r + 3d = (7r)^4
Now, to find the common difference (d), we need more information.
To find the common difference (d) of the arithmetic progression (AP), we need to use the information given:
1. The first, second, and fifth terms of the AP are three consecutive terms of an exponential sequence.
2. The first term of the linear sequence is 7.
Let's first determine the exponential sequence using the given information. Let's assume the exponential sequence is of the form a, ar, ar^2, where a is the first term and r is the common ratio.
Given that the first term of the linear sequence is 7, we can equate it to the first term of the exponential sequence. So, we have a = 7.
Since the first, second, and fifth terms of the AP are three consecutive terms of the exponential sequence, we can set up the following equations:
First term of AP (a) = First term of exponential sequence (a)
Second term of AP = Second term of exponential sequence (ar)
Fifth term of AP = Third term of exponential sequence (ar^2)
Using the values we have, we can write the following equations:
a = 7
ar = 3rd term of exponential sequence
ar^2 = 5th term of exponential sequence
Now, we need to find the common ratio (r) to determine the exponential sequence.
To find the common ratio, let's consider the ratio between the second and first terms of the exponential sequence:
(ar) / (a) = r
We can substitute the value of the first term (a = 7) into the equation:
(ar) / (7) = r
Since the second term of the AP is the second term of the exponential sequence, we can substitute the value of the second term (ar) into the equation. Let's assume it's x:
(x) / (7) = r
Now, we can set up a similar equation using the fifth term of the exponential sequence:
(ar^2) / (ar) = r
Substituting the value of the first term (a = 7) into the equation:
(7r^2) / (x) = r
Since the fifth term of the AP is the third term of the exponential sequence, we can substitute the value of the third term (ar^2) into the equation. Let's assume it's y:
(7r^2) / (y) = r
Now, we have two equations involving r:
(x) / (7) = r
(7r^2) / (y) = r
Rearranging the equations:
x = 7r
7r^2 = ry
Since the first, second, and fifth terms of the AP have been assumed to be consecutive terms of the exponential sequence, we can equate them with the corresponding terms:
First term of AP (a) = a = 7
Second term of AP = ar = x
Fifth term of AP = ar^2 = y
We can substitute these values into the equations above:
x = 7r
7r^2 = ry
Since x = 7r, we can substitute it in the second equation:
7r^2 = r(7r)
Simplifying:
7r^2 = 7r^2
This equation shows that r can have any value, as both sides are equal.
Therefore, the common difference (d) of the arithmetic progression is not determined uniquely. The given information is insufficient to determine the value of the common difference.