The geometric mean between the first two terms in a geometric sequence is 32. if the third term is 4 find the first term
Forget the previous post, I didn't read it carefully
let the 3 terms be a, ar , ar^2
geometric mean of first two terms = √(a(ar)) = 32
a √r = 32 --> a = 32/√r
also ar^2 = 4 --> a = 4/r^2
32/√r = 4/r^2
32r^2 = 4√r
8r^2 = √r
64r^3 = r
r^2 = 1/64
r = ± 1/8
from a = 4/r^2
a = 4(1/64) = 256
Ty
let the terms be a, ar, 4
4a = 32
a = 8 <---- done here
Bonus marks:
ar^2 = 4
8r^2 = 4
r^2 = 1/2
r = ± 1/√2
terms would be
8 , 8/√2, 8/√2^2
8 , 8/√2, 4
looks good,
a = 8
To find the first term in a geometric sequence, we can use the formula for the geometric mean:
Geometric mean = √(first term * second term)
Let's denote the first term as "a" and the second term as "b". We are given that the geometric mean between the first two terms is 32, so we can write this as:
32 = √(a * b)
We are also given that the third term in the sequence is 4, so we can represent this as:
b = 4
Substituting this value into the geometric mean equation, we have:
32 = √(a * 4)
Squaring both sides of the equation to eliminate the square root, we get:
1024 = a * 4
Dividing both sides by 4, we find:
256 = a
Therefore, the first term in the geometric sequence is 256.