The 3rd and 5th term of a g.p is 9 and 16.find the 4th term
ar^2 = 9
ar^4 = 16
divide them:
r^2 = 16/9
r = ± 4/3
then a(16/9) = 9
a = 81/16
term(4) = ar^3 = (81/16)(±4/3)^3 = ± 12
or , in a simple way ....
once we have r, just multiply by term(3) to get to term(4)
9*(±4/3) = ±12
To find the 4th term of a geometric progression (g.p.), we can use the formula:
nth term = a * r^(n-1),
where:
- a is the first term of the g.p.,
- r is the common ratio, and
- n is the position of the term.
Given that the 3rd term is 9 and the 5th term is 16, we can set up two equations to find the values of a and r.
Equation 1: a * r^2 = 9 (As the 3rd term is 9, n = 3)
Equation 2: a * r^4 = 16 (As the 5th term is 16, n = 5)
Now we can solve these equations simultaneously to find the values of a and r.
To find the 4th term of a geometric progression (g.p.), we need to determine the common ratio (r) first.
Given that the 3rd term (a3) is 9 and the 5th term (a5) is 16, we can use these values to find the common ratio.
The formula to find any term (an) of a geometric progression is:
an = a1 * r^(n-1)
Using this formula, we can set up two equations:
a3 = a1 * r^2 (since n = 3)
a5 = a1 * r^4 (since n = 5)
Substituting the given values:
9 = a1 * r^2 (Equation 1)
16 = a1 * r^4 (Equation 2)
Now, divide Equation 2 by Equation 1 to eliminate a1:
16/9 = (a1 * r^4)/(a1 * r^2)
16/9 = r^2
Taking the square root of both sides:
√(16/9) = r
r = 4/3 or -4/3
Since the common ratio cannot be negative in this case, we can conclude that r = 4/3.
Now, we can find the 4th term (a4) using the formula:
a4 = a1 * r^(4-1)
a4 = a1 * r^3
To find a1, we can use the information from a3:
9 = a1 * (4/3)^2
9 = a1 * (16/9)
a1 = (9 * 9)/16
a1 = 81/16
Now, substitute the values we found back into the equation for a4:
a4 = (81/16) * (4/3)^3
Calculating this expression gives us the 4th term of the geometric progression.