If 3,A,B,192 are consecutive terms of a gp . find the values of A and B
If3,a,b192 are consecutive term of g. P find the value of a and b
Tn = ar^n-1
3r^4-1 = 192
3r^3=192
devide both side by 3
r^3=64
r = 3 square root of 64
r=4
Pls solve it
To find the values of A and B in a geometric progression (GP), we can use the formula for the nth term of a GP:
Term(n) = a * r^(n-1)
where Term(n) represents the nth term of the GP, a is the first term, r is the common ratio, and n is the position of the term.
In this case, we are given that the terms are 3, A, B, and 192. Since they are consecutive terms, we know that A is the second term and B is the third term.
Step 1: Determine the common ratio (r)
We can find the common ratio by dividing any term in the sequence by its previous term. Let's use the formula for the second term to find the common ratio:
A = 3 * r^(2-1)
A = 3 * r
Step 2: Determine A
Since A is the second term, we can substitute it back into the formula to solve for A:
A = 3 * r
A = 3 * (B / A)
A^2 = 3B
Step 3: Determine B
We know that B is the third term, so we can substitute it into the formula as well:
B = 3 * r^(3-1)
B = 3 * r^2
Step 4: Substitute A and B equations
Substituting the equation for A (from Step 2) into the equation for B (Step 3) gives us:
B = 3 * (3B) / A^2
B = 9B / A^2
Cross-multiplying, we get:
B * A^2 = 9B
Simplifying, we have:
A^2 = 9
Taking the square root of both sides, we find:
A = ±3
Substituting A = 3 into the equation for B, we have:
B = 3 * r^2
B = 3 * (B / A)
B = 3 * (B / 3)
B = B
This equation implies that B can have any value, as long as it is not zero.
Therefore, the possible values for A are 3 and -3, and the value of B can be any non-zero number.
3 r^3 = 192 ... r^3 = 64
A = 3 r
B = 3 r^2
find r and solve