The third term of a G.p is 9 and the fifth term is 16. Find the 4th term.
The thirt term of a go is 9 and the fifth term is 16 find the 4th term
The third term of a G P is 9 and the fifth term is 16 find the 4th term
r^2 = 16/9
r = ±4/3
9 * ±4/3 = ±12
Mr ooblek pls that is the wrong working But still the same answer
Fourth =12
Well, well, well, we've got ourselves a geometric progression puzzle! To find the fourth term, we need a little math magic. Let me put on my clown hat and get to work!
In a geometric progression (or G.P.), each term is found by multiplying the previous term by a common ratio. Let's call that ratio "r".
So, we have the third term as 9, which means that the second term must be 9/r, and the first term must be (9/r)/r = 9/r².
Similarly, the fifth term is 16, which means that the fourth term is 16/r.
Now, we can set up an equation based on the G.P. formula:
16/r = (9/r²) * r
To solve this equation, we can simplify it by multiplying both sides by r²:
16r² = 9r
Now, let me tell you a little math joke to lighten things up:
Why don't mathematicians need to solve problems in the dark?
Because they have their own bright ideas!
Alright, back to the equation. We can rearrange it:
16r² - 9r = 0
Now, factor out an r:
r(16r - 9) = 0
To find the value of r, we set each factor equal to zero:
r = 0 or 16r - 9 = 0
Now, we can solve for r:
r = 0, or r = 9/16.
Since r can't be zero (otherwise, we'll have a G.P. of only zeros), we choose r = 9/16.
Therefore, the fourth term of the G.P. is given by:
Fourth term = 16/r = 16/(9/16) = 256/9.
So, my dear friend, the fourth term of the G.P. is 256/9.
To find the fourth term of a geometric progression (G.P), we need to determine the common ratio.
In a G.P, each term (except the first) is obtained by multiplying the previous term by a constant called the common ratio. Let's denote the common ratio as 'r'.
Given that the third term is 9 and the fifth term is 16, we can set up the following equations:
Third term: a * r^2 = 9 ---(1)
Fifth term: a * r^4 = 16 ---(2)
Here, 'a' represents the first term of the G.P.
To find 'r', divide equation (2) by equation (1):
(a * r^4) / (a * r^2) = 16 / 9
r^2 = 16 / 9
Taking the square root of both sides, we get:
r = √(16 / 9)
r = 4 / 3
Now, we can find the first term 'a' by substituting the value of 'r' into Equation (1):
a * (4 / 3)^2 = 9
a * (16 / 9) = 9
16a = 81
a = 81 / 16
Thus, the first term of the sequence is a = 81 / 16.
Finally, we can find the fourth term by substituting the values of 'a' and 'r' into the equation for the G.P formula:
Fourth term = a * r^3
Substituting the values:
Fourth term = (81 / 16) * (4 / 3)^3
Fourth term = (81 / 16) * (64 / 27)
Fourth term = 81 * 64 / (16 * 3)
Fourth term = 5184 / 48
Fourth term = 108
Therefore, the fourth term of the G.P is 108.