1.Given a geometric sequence with a_1=6 and r=2/3, write an explicit formula for a_n, the nth term of the sequence.
2.A geometric sequence has a_4=4 and a_5=7. What is a?
an = a1 r^(n-1) = 6 * (2/3)^(n-1)
= 6 * 2(n-1) /3^(n-1)
Plug and chug:
http://www.mathsisfun.com/algebra/sequences-sums-geometric.html
ok thanks i get #2 but i still don't understand 1.
1. To find the explicit formula for the nth term, we can use the formula for a geometric sequence:
aₙ = a₁ * r^(n-1),
where aₙ is the nth term, a₁ is the first term, r is the common ratio, and n is the position of the term.
In this case, a₁ = 6 and r = 2/3. Substituting these values into the formula, we get:
aₙ = 6 * (2/3)^(n-1).
Therefore, the explicit formula for the nth term of the sequence is aₙ = 6 * (2/3)^(n-1).
2. We are given a₄ = 4 and a₅ = 7 for a geometric sequence. From the formula for a geometric sequence, we have:
aₙ = a₁ * r^(n-1),
where aₙ is the nth term, a₁ is the first term, r is the common ratio, and n is the position of the term.
We can use the given values to form two equations:
a₄ = a₁ * r^(4-1) = a₁ * r³ = 4 (Equation 1)
a₅ = a₁ * r^(5-1) = a₁ * r⁴ = 7 (Equation 2)
To solve these equations, we can divide Equation 2 by Equation 1:
(a₁ * r⁴) / (a₁ * r³) = 7 / 4
r = 7/4
Now, we can substitute the value of r back into either Equation 1 or Equation 2 to find the value of a, the first term. Let's use Equation 1:
a₁ * (7/4)³ = 4
Simplifying the equation:
a₁ * (343/64) = 4
a₁ = (4 * 64) / 343
a₁ = 256 / 343
Therefore, a = a₁ = 256/343.