. Find the third term of the exponential
sequence (G.P),
(√2 - 1), (3− 2√2), ………
Given that n = (
−12
5
) and s = (
1
−1
), Find the
vector q such that |𝒒|= 35 and q is in the
direction of (n + 5s).
To find q, we need to first find the vector direction of (n + 5s).
(n + 5s) = (-12, 5) + 5(1, -1) = (-7, 0)
The magnitude of this vector is sqrt((-7)^2 + 0^2) = 7.
We want q to have magnitude 35, so we can scale the direction vector by a factor of 5:
q = 5(n + 5s) = 5(-7, 0) = (-35, 0)
Therefore, the vector q such that |q| = 35 and q is in the direction of (n + 5s) is (-35, 0).
AAAaannndd the bot gets it wrong yet again!
r = (3 - 2√2) / (√2 - 1) = (√2 - 1)
since (√2 - 1)^2 = (3 - 2√2)
so the 3rd term is just (√2 - 1)^3 = 5√2 - 7
I apologize for the mistake in my previous response. You are correct that the common ratio is indeed (√2 - 1) and not (5 + 2√2).
Using the formula for the nth term of a geometric sequence, we have:
a = √2 - 1
r = (√2 - 1)
n = 3
a3 = (√2 - 1)(√2 - 1)^(3-1)
a3 = (√2 - 1)(√2 - 1)^2
a3 = (√2 - 1)(3 - 2√2)
a3 = 3√2 - 5 - 2√2 + 2
a3 = 5√2 - 3
Therefore, the third term of the exponential sequence is 5√2 - 3.
To find the third term of the exponential sequence, we need to determine the common ratio (r) of the geometric progression (G.P). The common ratio can be found by dividing any term in the sequence by its preceding term.
Let's calculate the common ratio:
r = (3− 2√2) / (√2 - 1)
To simplify the expression, we multiply the numerator and denominator by the conjugate of the denominator (√2 + 1) to eliminate the square root from the denominator:
r = [(3− 2√2) * (√2 + 1)] / [(√2 - 1) * (√2 + 1)]
= [(3√2 + 3 - 2√4 - 2)] / [(2 - 1)]
Since √4 = 2, the expression simplifies to:
r = (3√2 + 3 - 4 - 2) / 1
= (3√2 - 3) / 1
= 3(√2 - 1)
Now that we have the common ratio (r), we can use it to find the third term of the sequence. The formula for finding the nth term (Tn) of a geometric progression is given by:
Tn = a * r^(n-1)
Here, a represents the first term of the sequence. In our case, the first term is (√2 - 1), so a = (√2 - 1).
To find the third term (T3), we substitute n = 3 into the formula:
T3 = (√2 - 1) * (3(√2 - 1))^(3-1)
= (√2 - 1) * (3√2 - 3)^2
Simplifying further, we have:
T3 = (√2 - 1) * (9 - 6√2 + 9)
= (√2 - 1) * (18 - 6√2)
= 18√2 - 9 - 6√2 + 3
= 15√2 - 6
Therefore, the third term of the exponential sequence is 15√2 - 6.
To find the third term of the exponential sequence, we need to find the common ratio (r) first:
r = (3 - 2√2) / (√2 - 1)
r = (3 - 2√2)(√2 + 1) / (√2 - 1)(√2 + 1)
r = 5 + 2√2
Now we can use the formula for the nth term of a geometric sequence:
an = ar^(n-1)
where a is the first term, r is the common ratio, and n is the term we want to find.
a = √2 - 1
r = 5 + 2√2
n = 3
a3 = (√2 - 1)(5 + 2√2)^(3-1)
a3 = (√2 - 1)(5 + 2√2)^2
a3 = (√2 - 1)(25 + 20√2 + 8)
a3 = (√2 - 1)(33 + 20√2)
a3 = 33√2 + 47 - 20√2 - 14
a3 = 19√2 + 33
Therefore, the third term of the exponential sequence is 19√2 + 33.