If 5th term of an exponential sequence is greater than the 4th term by thirteen and half and the 4th term is greater than the 3rd term by 9,find the common ratio and the 1st term
Xn = a r^(n-1)
X5 = a r^4
X4 = a r^3
ar^4 -ar^3 = 13.5
X3=a r^2
ar^3-ar^2 = 9 ---> ar^4-ar^3 = 9r
so
13.5 = 9 r
I think you can take it from there.
NO
To find the common ratio and the first term of an exponential sequence, we can use the formula for the nth term of an exponential sequence:
nth term = a * r^(n-1)
where:
- nth term is the value of the term we want to find
- a is the first term of the sequence
- r is the common ratio
- n is the position of the term in the sequence
Let's solve the problem step by step.
We are given two pieces of information:
1) The 4th term is greater than the 3rd term by 9.
2) The 5th term is greater than the 4th term by 13.5.
Let's assign variables to these terms:
- Let the 3rd term be A.
- Let the 4th term be B.
- Let the 5th term be C.
According to the given information:
B = A + 9 (equation 1)
C = B + 13.5 (equation 2)
To find the common ratio (r), we can divide the 4th term by the 3rd term:
r = B / A
Using equation 1, we can substitute B in terms of A:
r = (A + 9) / A
Now, let's substitute this value of r into equation 2 to get an equation in terms of A and C:
C = (A + 9) / A + 13.5
We know that C is the 5th term. So, the 5th term can be written in terms of the first term (a) and the common ratio (r):
C = a * r^(5-1)
C = a * r^4
Now, let's substitute r = (A + 9) / A into this equation:
C = a * ((A + 9) / A)^4
Therefore, we have two equations: equation 3 (from substituting B in equation 1) and equation 4 (from substituting C in the nth term formula).
1) B = A + 9 (equation 3)
2) C = a * ((A + 9) / A)^4 (equation 4)
Now, let's solve these equations to find A (the first term) and C (the 5th term).
From equation 3: B = A + 9, we can substitute B in terms of A in equation 4:
C = a * ((B - 9) / A)^4 (by substituting B = A + 9)
Now, we have C in terms of A and a only.
Let's say we have a value for the first term (a), and we want to find the common ratio (r) and the 5th term (C).
We can solve equation 4 for C using this value of a, and then substitute the value of C into equation 3 to find A. From A and C, we can then find the common ratio (r = (A + 9) / A).
Please provide the value of the first term (a) to continue with the calculation.