1. The 5th, 9thand16th terms of a linear sequence are consecutive terms of an exponential sequence
GP. Find the common difference in terms of the first term�
(a+8d)/(a+4d) = (a+15d)/(a+8d)
(a+8d)(a+8d) = (a+4d)(a+15d)
a^2+16ad+64d^2 = a^2+19ad+60d^2
4d^2 = 3ad
4d = 3a
d = 3a/4
I do not still understand
To find the common difference in terms of the first term in the linear sequence, we need to look at the given terms: the 5th, 9th, and 16th.
Let's assume the first term of the linear sequence is 'a' and the common difference is 'd'.
The 5th term of the linear sequence is given by:
a + 4d
The 9th term of the linear sequence is given by:
a + 8d
The 16th term of the linear sequence is given by:
a + 15d
According to the given information, these terms are consecutive terms of an exponential sequence. In an exponential sequence, consecutive terms have a common ratio.
So, we can set up the following equation based on the exponential sequence:
(a + 8d)/(a + 4d) = (a + 15d)/(a + 8d)
Now, let's solve this equation step-by-step:
1. Cross-multiply:
(a + 8d)(a + 8d) = (a + 4d)(a + 15d)
2. Expand both sides:
a² + 16ad + 64d² = a² + 19ad + 60d²
3. Simplify by cancelling out the common terms:
16ad + 64d² = 19ad + 60d²
4. Rearrange the equation by moving all the terms to one side:
16ad - 19ad = 60d² - 64d²
5. Simplify further:
-3ad = -4d²
6. Divide both sides of the equation by -d:
3a = 4d
Finally, we can express the common difference (d) in terms of the first term (a) as:
d = (3/4)a
Therefore, the common difference in terms of the first term in the linear sequence is (3/4)a.
To find the common difference of the linear sequence in terms of the first term, we need to compare the differences between consecutive terms in the linear sequence.
Let's start by writing the formulas for the terms of the linear sequence and the exponential sequence:
For the linear sequence:
a + (n - 1)d, where "a" is the first term and "d" is the common difference.
For the exponential sequence:
ar^(n - 1), where "a" is the first term and "r" is the common ratio.
Given that the 5th, 9th, and 16th terms of the linear sequence are consecutive terms of the exponential sequence, we can set up the following equations:
a + (5 - 1)d = ar^(9 - 1)
a + (9 - 1)d = ar^(16 - 1)
We can simplify the equations as follows:
a + 4d = ar^8
a + 8d = ar^15
Now, let's eliminate "a" from the equations. We can do this by subtracting the first equation from the second:
(a + 8d) - (a + 4d) = (ar^15) - (ar^8)
4d = ar^15 - ar^8
Next, we can factor out "a" from the right side of the equation:
4d = a(r^15 - r^8)
Since we want to express the common difference "d" in terms of the first term "a," we can divide both sides of the equation by "4" and then divide both sides by (r^15 - r^8):
d = a(r^15 - r^8) / 4(r^15 - r^8)
d = a / 4
Therefore, the common difference of the linear sequence in terms of the first term "a" is a / 4.