The 5th,9th and 16th terms of a linear sequence A.P are consecutive terms of an exponential sequence.Find the common difference of the linear sequence in terms of the first term.
since the GP has a common ratio,
(a+8d)/(a+4d) = (a+15d)/(a+8d)
(a+8d)^2 = (a+15d)(a+4d)
a^2+16ad+64d^2 = a^2+19ad+60d^2
3ad=4d^2
3a = 4d
Wrong ans
Well, it seems like we have a sequence going on a never-ending roller coaster ride! Let's see if we can make sense of it.
We know that the 5th, 9th, and 16th terms of the linear sequence are also consecutive terms of an exponential sequence. That's quite an unconventional meetup, don't you think?
To find the common difference of the linear sequence, let's assume that the first term of the linear sequence is 'a', and the common difference is 'd'.
Since the 5th term is consecutive with the 9th term, we can say:
a + 4d = a + 8d
Simplifying this, we get:
4d = 8d
Hmm, it seems like our roller coaster is stuck in a loop! This equation suggests that the common difference of the linear sequence is actually zero!
So, in terms of the first term 'a', the common difference 'd' of the linear sequence is 0. Which means the sequence is actually just a series of identical terms. Talk about a disappointing ride!
I hope this helps and brings a smile to your face, even if the sequence itself isn't so thrilling!
To find the common difference of the linear sequence in terms of the first term, we need to understand the given information and use it to find a relationship between the terms.
Let's break down the given information:
1. The 5th term of the linear sequence is a consecutive term of an exponential sequence.
2. The 9th term of the linear sequence is also a consecutive term of the same exponential sequence.
3. The 16th term of the linear sequence is a consecutive term of the exponential sequence as well.
To find the relationship between terms in an exponential sequence, we know that each term is obtained by multiplying the previous term by a constant factor. Let's say this constant factor is denoted by 'r.'
Now, let's solve the problem step by step:
Step 1: Considering the linear sequence, the general term of an arithmetic progression (AP) is given by:
a + (n-1)d
where 'a' is the first term and 'd' is the common difference.
Step 2: Let's consider the 5th, 9th, and 16th terms of the linear sequence:
Term 5 = a + 4d
Term 9 = a + 8d
Term 16 = a + 15d
Step 3: As mentioned earlier, all three of these terms are consecutive terms of an exponential sequence. In an exponential sequence, each term (except the first term) is obtained by multiplying the previous term by a constant factor. Let's denote this constant factor by 'r.'
Using this information, we can write the relationship between the exponential terms as:
(a + 8d) = r * (a + 4d)
(a + 15d) = r * (a + 8d)
Step 4: Now, we have two equations (equations 1 and 2) with two unknowns ('a' and 'd'). We can solve these equations to find the common difference in terms of the first term.
(1) a + 8d = r * (a + 4d)
(2) a + 15d = r * (a + 8d)
Solving these equations will give us the values of 'a' and 'd,' which will allow us to find the common difference in terms of the first term.