The 5th,9th and 16th terms of a linear sequence (A.P) are the consecutive terms of an exponential sequence (G.P)
(i) find the common difference of the linear sequence in terms of the first term
(ii)show that the 21st,37th and 65th terms of the linear sequence are also in an exponential sequence and hence find the common ratio
Pls help me solve
for the AP,
term5 = a+4d
term9 = a+8d
term16 = a + 15d
but these are now in a GP
(a+8d)/(a+4d) = (a+15d)/(a+8d)
(a+8d)^2 = (a+4d)(a+15d)
a^2 + 16ad + 64d^2 = a^2 + 19ad + 60d^2
3ad - 4d^2 = 0
d(3a - 4d) = 0
d = 0 which does not give an interesting AP
or
d = 3a/4
repeat with:
(a+36d)^2 = (a+20d)(a+64d)
a^2 + 72ad +1296d^2 = a^2 + 84ad + 1280d^2
16d^2 - 12ad = 0
4d^2 - 3ad = 0
d = 3a/4 , same result
r = term9/term5 = (a+8d)/(a+4d)
= (a + 8(3a/4))/(a + 4(3a/4) = 7a/4a = 7/4
just use your formulas.
(a+8d)/(a+4d) = (a+15d)/(a+8d)
3a = 4d
d = 3/4 a
Now show that the other three terms also have a common ratio
To solve this problem, we first need to understand some definitions:
Linear Sequence (Arithmetic Progression or A.P.):
A linear sequence is a sequence in which the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by 'd'.
Exponential Sequence (Geometric Progression or G.P.):
An exponential sequence is a sequence in which each term is obtained by multiplying the previous term by a constant number. This constant number is called the common ratio, denoted by 'r'.
Given Information:
The 5th, 9th, and 16th terms of a linear sequence are consecutive terms of an exponential sequence (G.P.).
Now let's solve the problem step by step:
(i) Finding the common difference of the linear sequence:
We are given that the 5th, 9th, and 16th terms of the linear sequence are consecutive terms of a geometric sequence (G.P.).
Let's assume the first term of the linear sequence is 'a', and the common difference is 'd'.
To find the 5th term:
5th term = a + 4d (since the common difference is d)
To find the 9th term:
9th term = a + 8d
To find the 16th term:
16th term = a + 15d
We are also given that these terms form a geometric sequence. In a geometric sequence, the ratio between consecutive terms is constant.
For the given information, we can write:
(a + 8d) / (a + 4d) = (a + 15d) / (a + 8d)
We can solve this equation to find the value of the common difference (d) in terms of the first term (a).
(ii) Showing that the 21st, 37th, and 65th terms of the linear sequence are also in a geometric sequence and finding the common ratio:
Now, using the value of the common difference (d) obtained in step (i), we can find the 21st, 37th, and 65th terms of the linear sequence.
The 21st term:
21st term = a + 20d
The 37th term:
37th term = a + 36d
The 65th term:
65th term = a + 64d
We need to show that these terms also form a geometric sequence.
For the given information, we can write:
(a + 36d) / (a + 20d) = (a + 64d) / (a + 36d)
We can solve this equation to find the value of the common ratio (r) in terms of the first term (a) and common difference (d).
Once we obtain the values for the common difference (in part i) and the common ratio (in part ii), we will have fully solved the problem.