the 4th term of a geometric sequence is 1/2 and the tenth term is 1/128 find the 10th term?
well i think you got the question wrong anyway
the tenth term is 1/128
the first term is 4
the "r" is 0.5
This question is right
To find the 10th term of a geometric sequence, we need to determine the common ratio of the sequence. The common ratio (r) is found by dividing any term by its preceding term.
Let's start by finding the common ratio using the 4th term:
4th term = 1/2
3rd term = (1/2) / r
2nd term = ((1/2) / r) / r
1st term = (((1/2) / r) / r) / r
Similarly, using the 10th term:
10th term = 1/128
9th term = (1/128) / r
8th term = ((1/128) / r) / r
...
2nd term = (((1/128) / r) / r) / r
1st term = ((((1/128) / r) / r) / r) / r
Since the 1st term and 2nd term in both cases represent the same term, we can equate them:
(((1/2) / r) / r) / r = ((((1/128) / r) / r) / r) / r
To solve for r, we can cross-multiply:
(((1/2) / r) / r) * r * r = (((1/128) / r) / r) / r
Canceling out the r's:
1/2 = 1/128 / r^3
To simplify further, we can multiply both sides by 128:
(1/2) * 128 = 1 / r^3
64 = 1 / r^3
Taking the cube root of both sides:
∛64 = ∛(1 / r^3)
4 = 1 / r
Multiplying both sides by r:
4r = 1
Dividing both sides by 4:
r = 1/4
Now that we have the common ratio (r = 1/4), we can find the 10th term by multiplying the 10th term by the common ratio raised to the power of 9 (since the 10th term is the 4th term raised to the power of (10 - 4) = 6):
10th term = (1/2) * (1/4)^6
Simplifying:
10th term = (1/2) * (1/4)^6
= (1/2) * (1/4096)
= 1/8192
Therefore, the 10th term of the geometric sequence is 1/8192.
To find the 10th term of a geometric sequence, we will first find the common ratio.
The 4th term of the sequence is 1/2. We can represent this as:
aᵢ = a₁ * r^(ᵢ-₁)
where aᵢ is the ith term, a₁ is the first term, and r is the common ratio.
Using this formula, we substitute aᵢ = 1/2 and ᵢ = 4:
1/2 = a₁ * r^(4-1)
1/2 = a₁ * r^3
Similarly, the 10th term of the sequence is 1/128. We can represent this as:
1/128 = a₁ * r^(10-1)
1/128 = a₁ * r^9
Now we have two equations:
1/2 = a₁ * r^3 - [Equation 1]
1/128 = a₁ * r^9 - [Equation 2]
Dividing Equation 2 by Equation 1, we can eliminate a₁:
(1/128) / (1/2) = (a₁ * r^9) / (a₁ * r^3)
1/128 ÷ 1/2 = r^9 ÷ r^3
1/128 × 2/1 = r^(9-3)
1/64 = r^6
To simplify further, we can take the sixth root of both sides:
(r^6)^(1/6) = (1/64)^(1/6)
r = 1/8
Now that we have found the common ratio (r = 1/8), we can substitute it back into Equation 2 to find the first term (a₁):
1/128 = a₁ * (1/8)^9
1/128 = a₁ * (1/8^9)
1/128 = a₁ * (1/8^9)
1/128 = a₁ * (1/134217728)
Simplifying further:
a₁ = 1/128 ÷ (1/134217728)
a₁ = 1/128 × 134217728/1
a₁ = 134217728/128
a₁ = 1048576
Now that we have the value of the first term (a₁ = 1048576) and the common ratio (r = 1/8), we can use the formula to find the 10th term:
a₁₀ = a₁ * r^(10-1)
a₁₀ = 1048576 * (1/8)^9
a₁₀ = 1048576 * (1/134217728)
Now, we simplify the expression:
a₁₀ = 1048576/134217728
a₁₀ = 1/128
Therefore, the 10th term of the geometric sequence is 1/128.