Find a formula for the nth term of the sequence
geometric
a1=4, ak+1 =1/2 ak
my answer
an=4(1/2)^n-1
Your answer is correct.
a₁ = 4
aₖ₊₁ = aₖ / 2
a₁₊₁ = a₂ = a₁ / 2 = 4 / 2 = 2
a₂₊₁ = a₃ = a₂ / 2 = 2 / 2 = 1
a₃₊₁ = a₄ = a₃ / 2 = 1 / 2
etc.
r = a₂ / a₁ = 2 / 4 = 1 / 2
r = a₃ / a₂ = 1 / 2
r = a₄ / a₃ = ( 1 / 2 ) / 1 = 1 / 2
In your geometric sequence, a = a₁ = 4 , r = 1 / 2
The n-th term of a geometric sequence:
aₙ = a ∙ r ⁿ ⁻ ¹
aₙ = 4 ( 1 / 2 )ⁿ ⁻ ¹
Also:
4=2^2
(1/2)^(n-1) =2^(-(n-1))=2^(-n+1)
an=4(1/2)^(n-1)=2^2*2^(-n+1)=2^2*2^(-n)*2^1=
2^2*2^1*2^(-n)=2^(2+1)*2^(-n)=2^3*2^(-n)=2^(3-n)
To find the formula for the nth term of a geometric sequence, you need to understand the pattern in the sequence. In a geometric sequence, each term is obtained by multiplying the previous term by a constant value known as the common ratio.
In this case, you are given that a1 (the first term) is 4, and each subsequent term (ak+1) is obtained by multiplying the previous term (ak) by 1/2.
To find the formula for the nth term, you can begin by looking at the first few terms of the sequence:
a1 = 4
a2 = (1/2) * a1 = (1/2) * 4 = 2
a3 = (1/2) * a2 = (1/2) * 2 = 1
a4 = (1/2) * a3 = (1/2) * 1 = 1/2
...
By observing these terms, you can see that each subsequent term is obtained by multiplying the previous term by 1/2. This means that the common ratio (r) in this geometric sequence is 1/2.
Now, to find the formula for the nth term, you can use the following formula for geometric sequences:
an = a1 * r^(n-1)
Plugging in the given values into the formula, you get:
an = 4 * (1/2)^(n-1)
Therefore, the formula for the nth term of the sequence is:
an = 4 * (1/2)^(n-1)