given 1,0.5,4,o.25,7,0.125,10
assume this pattern continues consistently
1.write down the 33th term in this sequence
2.determine the sum of the first 24 terms in this sequence.give the answer to five decimal place
2.1.in a geometric sequence the fourth term is root of 3 and the seventh term is root of 81 determine the first term and the constant ratio
#1
for n odd, Tn = 3n-2
For n even, Tn = 1/2^n
#2
r^3 = T7/T4 = √81/√3 = √27
r = √3
Now you can find a
To find the 33rd term in the given sequence, we can observe that the pattern alternates between multiplying by 2 and dividing by 2 each time.
Step 1: Start with the given numbers: 1, 0.5, 4, 0.25, 7, 0.125, 10.
Step 2: Multiply the previous term by 2 if the position is odd, and divide by 2 if the position is even.
- 1st term: 1 (odd position)
- 2nd term: 1 * 2 = 2 (even position)
- 3rd term: 2 (odd position)
- 4th term: 2 / 2 = 1 (even position)
Repeat this process until the desired term is found.
Continuing this pattern, we can determine the 33rd term as follows:
Step 3: Determine the parity of the position:
- 33 is an odd number, so the position is odd.
Step 4: Perform the required operations:
- 32nd term: 10 * 2 = 20
- 33rd term: 20 (odd position)
Therefore, the 33rd term in this sequence is 20.
To determine the sum of the first 24 terms in this sequence, we will use the formula for the sum of a geometric series:
S = a * (r^n - 1) / (r - 1),
where
S is the sum of the series,
a is the first term,
r is the common ratio, and
n is the number of terms.
Step 1: Identify the given numbers in the sequence:
- The first term (a) is 1.
- The common ratio (r) can be observed from the pattern: multiplying or dividing by 2.
- The number of terms (n) is 24.
Step 2: Plug in the values into the formula:
- a = 1
- r = 2
- n = 24
S = 1 * (2^24 - 1) / (2 - 1)
S = (2^24 - 1)
Step 3: Calculate the sum:
- Using a calculator or software, evaluate (2^24 - 1) to get the sum.
The result, to five decimal places, is the sum of the first 24 terms in the sequence.
Now moving on to the second question:
Given a geometric sequence with the fourth term as the square root of 3 and the seventh term as the square root of 81, we need to find the first term and the common ratio.
Step 1: Write down the given terms:
- Fourth term: sqrt(3)
- Seventh term: sqrt(81)
Step 2: Determine the positions of the terms:
- Fourth term: position 4
- Seventh term: position 7
Step 3: Use the formula for a geometric sequence term:
- For the nth term (Tn) in a geometric sequence, the formula is:
Tn = a * (r^(n-1)), where a is the first term and r is the common ratio.
Step 4: Write the equations using the given terms and positions:
- sqrt(3) = a * (r^(4 - 1))
- sqrt(81) = a * (r^(7 - 1))
Simplify the equations:
- Equation 1: sqrt(3) = a * (r^3)
- Equation 2: 9 = a * (r^6)
Step 5: Divide Equation 2 by Equation 1 to eliminate "a":
- 9 / sqrt(3) = a * (r^6) / (a * (r^3))
- 3 = r^3
Step 6: Solve for "r":
- Take the cube root of both sides: r = ∛3
Step 7: Substitute the value of "r" into Equation 1 to find "a":
- sqrt(3) = a * (∛3^3)
- sqrt(3) = a * 3
- Divide both sides by 3: a = sqrt(3) / 3
Therefore, the first term (a) is sqrt(3) / 3, and the common ratio (r) is ∛3.