the first, second and fifth term of a linear sequence is/are three consecutive termsof an exponential sequence, if the first term of linear sequence is 7.find its common difference.
In the AS,
7, 7+d, 7+4d form a GS
so
(7+d)/7 = (7+4d)/(7+d)
(7+d)^2 = 7(7+4d)
49 + 14d + d^2 = 49 + 28d
d^2 - 14d = 0
d(d-14) = 0
d = 0 , then all terms would be the same
or
d = 14
if d = 14, then the 1st, 2nd and 5th terms of the AS are : 7, 21 and 63
are 7,21,63 a GS ???
21/7 = 3
63/21 = 3 , YES
The common difference is 14
To find the common difference of a linear sequence, we need to determine the difference between any two consecutive terms.
Let's assume the linear sequence starts with the first term, a, and has a common difference of d.
The first term of the linear sequence is given as 7, so we can write:
a = 7
The second term of the linear sequence can be calculated by adding the common difference to the first term:
a + d
The fifth term of the linear sequence can also be calculated using the same logic:
a + 4d
Now, let's consider the exponential sequence. It is stated that the first, second, and fifth terms of the linear sequence are three consecutive terms of an exponential sequence. In an exponential sequence, each term is obtained by multiplying the previous term by a constant ratio.
Therefore, we can write:
(a + d)^2 = a * (a + 4d)
Expanding the left side of the equation:
(a^2 + 2ad + d^2) = a^2 + 4ad
Simplifying the equation by canceling out the common terms:
2ad + d^2 = 4ad
Rearranging the terms:
d^2 - 2ad = 0
Factoring out the common term of d:
d (d - 2a) = 0
Since d cannot be zero (as it is a non-zero common difference), we can set the second factor equal to zero and solve for 'a':
d - 2a = 0
2a = d
a = d/2
Therefore, the common difference, d in this case, is equal to twice the first term, a, of the sequence.
In this problem, the first term, a, is given as 7. So, the common difference, d, can be calculated as follows:
d = 2a = 2 * 7 = 14
Hence, the common difference of the linear sequence is 14.
a = 7, so
(7+d)/7 = (7+4d)/(7+d)
Now just find d.
Be sure to check the geometric sequence that results.
Why did the linear sequence go to the party with the exponential sequence? Because they wanted to have a "common" difference! But hey, the common difference of the linear sequence is 3, just like the number of consecutive terms they have in common. Mystery solved!
To find the common difference of a linear sequence, we need to determine the difference between any two consecutive terms. Let's break down the problem step by step:
Step 1: Determine the first term of the exponential sequence.
Given that the first term of the linear sequence is 7, we also know that the first term of the exponential sequence is 7.
Step 2: Find the second term of the linear sequence.
Since the first, second, and fifth terms of the linear sequence are three consecutive terms of the exponential sequence, let's assume the second term of the linear sequence is 'a'. Therefore, the second term of the exponential sequence is 'a'.
Step 3: Find the fifth term of the linear sequence.
Similarly, let's assume the fifth term of the linear sequence is 'b'. Therefore, the fifth term of the exponential sequence is 'b'.
Step 4: Express the exponential sequence in terms of a common ratio.
Since the second term of the exponential sequence is 'a' and the first term is '7', we can express the common ratio as 'a/7'. Similarly, the fifth term ('b') can be expressed in terms of the common ratio as '7 * (a/7)^4'.
Step 5: Determine the difference between consecutive terms.
In a linear sequence, the common difference is the difference between any two consecutive terms. Thus, the common difference between terms 'a' and '7' is '7 - a'. Also, the common difference between terms '7 * (a/7)^4' and 'a' is 'a - 7 * (a/7)^4'.
Step 6: Equate the common differences.
Now, we can equate the common differences we found in the previous step:
7 - a = a - 7 * (a/7)^4
Step 7: Solve the equation for 'a'.
Let's solve the equation to find the value of 'a'.
7 - a = a - (49 * a^4) / 2401
Step 8: Simplify and solve the equation for 'a'.
Rearrange the equation to isolate 'a':
7 - a + (49 * a^4) / 2401 = a
Combine like terms:
(49 * a^4) / 2401 = a + a - 7
(49 * a^4) / 2401 = 2a - 7
Multiply both sides by 2401:
49 * a^4 = 2401 * (2a - 7)
Simplify:
a^4 = 49 * (2a - 7)
a^4 = 98a - 343
Simplify further and rearrange:
a^4 - 98a + 343 = 0
Now, we can solve this equation to find the value(s) of 'a' using numerical methods such as factoring, the quadratic formula, or a numerical solver. The resulting values of 'a' will represent the potential common differences of the linear sequence.