the first term of gp is 64 & the common ratio is half . how many terms must be added to obtain a total of 127 and a half
what a load of rubbish.
The sequence goes 64, 32, 16, ... + 1
To find how many terms sum to 127, solve
64(1 - 1/2^n)/(1 - 1/2) = 127
n = 7
But then, we already knew that, since 64=2^6
Oops. My mistake. I did not see the 127 1/2
But, it's easy, since we already know that 7 terms sum to 127, and the next term is 1/2, so 8 terms sum to 127.5
No problem, thank you for pointing out the mistake. You are absolutely right. We already know that 7 terms sum to 127, and since the next term is 1/2, adding an additional term will give us the total of 127.5.
Hence, we need to add 8 terms in order to obtain a total of 127.5.
To find the number of terms needed to obtain a total of 127 and a half in a geometric progression (GP) with a first term of 64 and a common ratio of half, follow these steps:
Step 1: Calculate the sum of the GP using the formula for the sum of a finite geometric series:
Sum = (a * (r^n - 1)) / (r - 1)
where:
a = first term
r = common ratio
n = number of terms
Let's plug in the given values into the formula:
127.5 = (64 * (0.5^n - 1)) / (0.5 - 1)
Step 2: Simplify the equation:
127.5 = (64 * (0.5^n - 1)) / (-0.5)
Step 3: Multiply both sides of the equation by (-0.5) to get rid of the denominator:
127.5 * (-0.5) = 64 * (0.5^n - 1)
Step 4: Simplify the equation further:
-63.75 = 64 * (0.5^n - 1)
Step 5: Divide both sides of the equation by 64 to isolate the bracket:
(-63.75) / 64 = 0.5^n - 1
Step 6: Add 1 to both sides of the equation:
((-63.75) / 64) + 1 = 0.5^n
Step 7: Simplify the equation:
((-63.75) + 64) / 64 = 0.5^n
Step 8: Find the value of the bracketed expression:
0.25 / 64 = 0.5^n
Step 9: Rearrange the equation to solve for n:
0.25 = 64 * 0.5^n
Step 10: Divide both sides of the equation by 64:
0.25 / 64 = 0.5^n
Step 11: Simplify the equation:
0.00390625 = 0.5^n
Step 12: Take the logarithm (base 0.5) of both sides of the equation to solve for n:
log base 0.5 (0.00390625) = n
Step 13: Use a calculator to find the logarithm:
n ≈ log base 0.5 (0.00390625) ≈ 8
Therefore, you would need to add 8 terms to the GP to obtain a total of 127 and a half.
To find the number of terms needed to obtain a total of 127.5 in a geometric progression (GP) with a first term of 64 and a common ratio of 1/2, you can use the formula for the sum of a finite geometric series.
The sum of a finite geometric series is given by the formula:
S_n = a * (1 - r^n) / (1 - r)
Where:
S_n is the sum of the first n terms of the GP,
a is the first term of the GP,
r is the common ratio of the GP,
n is the number of terms.
In this case, the sum (S_n) is given as 127.5, the first term (a) is 64, and the common ratio (r) is 1/2. We need to solve for the number of terms (n).
Substituting the given values into the formula, we have:
127.5 = 64 * (1 - (1/2)^n) / (1 - (1/2))
Simplifying the equation further, we get:
127.5 = 128 * (1 - (1/2)^n)
Divide both sides of the equation by 128:
127.5 / 128 = 1 - (1/2)^n
Rewriting 127.5 as a fraction:
255/2^8 = 1 - (1/2)^n
Simplifying the right side of the equation, we get:
255/2^8 = (2^n - 1) / 2^n
Cross-multiplying the equation:
(255/2^8) * 2^n = 2^n - 1
Multiplying both sides by 2^8 to get rid of the denominator, we have:
255 * 2^n = 2^n - 1
Next, let's isolate the term with the variable (2^n) on the left side:
255 * 2^n - 2^n = 1
Now, combine the terms with 2^n on the left:
(255 - 1) * 2^n = 1
254 * 2^n = 1
Divide both sides by 254:
2^n = 1/254
To solve for n, we can take the logarithm of both sides:
n * log(2) = log(1/254)
Divide both sides by log(2):
n = log(1/254) / log(2)
Using a calculator to evaluate the logarithms:
n ≈ -7.980
Since the number of terms must be a positive whole number, we round up to the nearest whole number:
n = 8
Therefore, to obtain a total of 127.5, you would need to add 8 terms to the geometric progression.
To find the number of terms needed to obtain a total of 127.5, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r), where S is the desired sum, a is the first term, and r is the common ratio.
In this case, we want the sum to be 127.5, the first term is 64, and the common ratio is 1/2:
127.5 = 64 / (1 - 1/2)
Simplifying the equation:
127.5 = 64 / (1/2)
127.5 = 64 * 2
127.5 = 128
Since the sum is equal to 128, we don't need to add any terms to obtain a total of 127.5.
Apologies for the incorrect response. You are absolutely right.
To find the number of terms needed to obtain a total of 127.5, we can solve the equation:
64(1 - (1/2)^n) / (1 - 1/2) = 127.5
Simplifying the equation:
64(1 - (1/2)^n) = 255
1 - (1/2)^n = 255/64
(1/2)^n = 1 - 255/64
(1/2)^n = 9/64
To simplify the equation, we can express 9/64 as (3/8)^2:
(1/2)^n = (3/8)^2
Taking the logarithm of both sides:
n log(1/2) = 2 log(3/8)
n = 2 log(3/8) / log(1/2)
n ≈ 2 * (-0.415) / (-0.301)
n ≈ 2.486
Since n represents the number of terms in the sequence, we need to round up to the nearest whole number. Therefore, we need to add 3 terms in order to obtain a total of 127.5.