In a G.P the second and fourth term are 0.04 and respectively find the common ratio and first term
proofread your post and supply the missing information.
then r^2 = a4/a2
Well, it seems like this G.P. is playing a little game of hide and seek with us. Would you like me to put on my detective hat and try to solve this mystery for you?
To find the common ratio and first term in a geometric progression (G.P.), we can use the formula for the nth term of a G.P.:
nth_term = a * r^(n-1)
Given that the second term (n = 2) is 0.04 and the fourth term (n = 4) is 0.16, we can form two equations:
0.04 = a * r^(2-1) -- Equation 1
0.16 = a * r^(4-1) -- Equation 2
Now, let's solve these equations step-by-step:
Step 1: Solve Equation 1 for a:
Substitute n = 2 and nth_term = 0.04 into the formula:
0.04 = a * r^(2-1)
0.04 = a * r
Step 2: Solve Equation 2 for a:
Substitute n = 4 and nth_term = 0.16 into the formula:
0.16 = a * r^(4-1)
0.16 = a * r^3
Step 3: Divide Equation 2 by Equation 1 to eliminate 'a':
Divide Equation 2 by Equation 1:
(0.16)/(0.04) = (a * r^3)/(a * r)
4 = r^2
Step 4: Take the square root of both sides:
Take the square root of both sides to solve for 'r':
√(4) = √(r^2)
2 = r
So, the common ratio (r) is 2.
Step 5: Substitute the value of 'r' into Equation 1 to solve for 'a':
Substitute r = 2 and Equation 1 into the formula:
0.04 = a * 2
0.04 = 2a
Step 6: Solve for 'a':
Divide both sides of the equation by 2 to solve for 'a':
0.04/2 = 2a/2
0.02 = a
So, the first term (a) is 0.02.
Therefore, the common ratio (r) is 2 and the first term (a) is 0.02 in the geometric progression.
To find the common ratio and first term of a geometric progression (G.P.) when the second and fourth terms are given, we can use the following steps:
Step 1: Understand the formula of a G.P.
The formula for the nth term of a geometric progression is:
an = a1 * r^(n-1)
where:
an = nth term
a1 = first term
r = common ratio
n = term number
Step 2: Use the given information to set up equations.
Let's set up equations using the given information:
a2 = 0.04 (second term)
a4 = ? (fourth term)
Step 3: Substitute the relevant values into the formula.
Using the formula for the second term, we have:
a2 = a1 * r^(2-1) = a1 * r
Step 4: Solve for the common ratio (r).
We can rearrange the equation to solve for r:
r = a2 / a1
Given that a2 = 0.04, we need to know the value of a1 in order to calculate r.
Step 5: Use the given fourth term to solve for a1.
Let's use the fourth term (a4) to find a1. Using the formula:
a4 = a1 * r^(4-1) = a1 * r^3
Step 6: Substitute the values into the equation.
Given that a4 = 0.04, we have:
0.04 = a1 * r^3
Step 7: Solve for the first term (a1).
We can rearrange the equation to solve for a1:
a1 = 0.04 / r^3
Now, if you have the value of r (from step 4), you can substitute it into the equation to find the value of a1.
Following these steps, you can determine both the common ratio (r) and the first term (a1) of the geometric progression.