The second and forth term of an exponential sequence (G .P) of positive term are 9 and 4 respectively. Find the
A. Common ratio
B. First term
C. Sum of the first terms of the sequence
I don't I understood
Please make it more explanatory
Men
To find the common ratio and first term of the exponential sequence, we can use the formula for the nth term of a geometric progression (G.P):
T_n = a * r^(n-1)
where T_n is the nth term, a is the first term, r is the common ratio, and n is the position of the term.
Given that the second term (T_2) is 9 and the fourth term (T_4) is 4, we can set up two equations:
T_2 = a * r^(2-1) = a * r
T_4 = a * r^(4-1) = a * r^3
We have two equations and two unknowns (a and r), so we can solve this system of equations.
From the first equation:
9 = a * r
From the second equation:
4 = a * r^3
To find the common ratio (r), we can divide the equations:
(9 / 4) = (a * r) / (a * r^3)
(9 / 4) = 1 / r^2
Cross-multiplying:
9r^2 = 4
r^2 = 4/9
r = √(4/9) or r = -√(4/9) (Since the terms are positive, we take the positive square root)
Thus, the common ratio (r) is 2/3.
To find the first term (a), we can use either equation. Let's use the first one:
9 = a * (2/3)
a = 9 * (3/2)
a = 27/2
Thus, the first term (a) is 27/2.
Now, to find the sum of the first terms of the sequence, we can use the formula for the sum of a geometric progression (G.P):
S_n = a * (1 - r^n) / (1 - r)
where S_n is the sum of the first n terms.
Since we want the sum of the first terms of the sequence, we can use this formula with n = 4 (since we are given the fourth term):
S_4 = (27/2) * (1 - (2/3)^4) / (1 - (2/3))
Now, you can simplify the expression to find the sum of the first terms of the sequence.
A. 9 r^2 = 4 ... r = √(4/9) = 2/3
B. f * 2/3 = 9 ... f = 27/2
C. how many terms?