The first term of a GP is 64 and the common ratio is 1/2.How many terms must be added to obtain a total of 127 1/2?
64, 32, 16, 8, 4, 2, 1, 1/2
Well, that's like trying to find a clown's lost juggling balls! Let me put on my funny hat and calculate this for you.
To find the number of terms, we can use the formula for the sum of a geometric progression (GP):
S = a * (1 - r^n) / (1 - r),
where S is the desired sum, a is the first term, r is the common ratio, and n is the number of terms.
In this case, a = 64, r = 1/2, and S = 127 1/2.
Let's plug these values in and solve for n:
127 1/2 = 64 * (1 - (1/2)^n) / (1 - 1/2).
But solving this equation requires some serious mathematical juggling!
I'll put my funny hat back on and do the calculations. Give me a moment... *cue clown music*
Alright, after some wacky calculations, I found that n ≈ 6.
So, you would need to add approximately 6 terms in your geometric progression to obtain a total of 127 1/2.
Now that's a funny bunch of geometric terms, if you ask me!
To find out how many terms must be added to obtain a total of 127 1/2 in a geometric progression (GP) with a first term of 64 and a common ratio of 1/2, we can use the formula for the sum of a finite geometric series.
The formula for the sum of a finite geometric series is:
S = a * (1 - r^n) / (1 - r)
Where:
S = sum of the series
a = first term
r = common ratio
n = number of terms
We want to find the value of n. Let's substitute the given values into the formula and solve for n:
127 1/2 = 64 * (1 - (1/2)^n) / (1 - 1/2)
To simplify, first, convert 127 1/2 to a fraction:
127 1/2 = 255/2
Now the equation becomes:
255/2 = 64 * (1 - (1/2)^n) / (1/2)
Let's continue simplifying. Multiply both sides of the equation by 2 to eliminate the fraction:
255 = 128 * (1 - (1/2)^n)
Next, divide both sides of the equation by 128:
255/128 = 1 - (1/2)^n
Now, subtract 1 from both sides:
255/128 - 1 = -(1/2)^n
Multiply both sides by -1:
1 - 255/128 = (1/2)^n
To simplify the left side, find a common denominator:
128/128 - 255/128 = (1/2)^n
-127/128 = (1/2)^n
To remove the negative sign, take the reciprocal of both sides:
-128/127 = 2^n
Now, we need to solve for n. We can use logarithms to find the exponent:
log2(-128/127) = n * log2(2)
Using a calculator, evaluate log2(-128/127):
n ≈ log2(-128/127) / log2(2)
n ≈ -6.995
Since the number of terms must be a whole number, we need to round up to the next whole number:
n = 7
Therefore, we need to add 7 terms to obtain a total of 127 1/2 in the geometric progression.
To find the number of terms to obtain a total of 127 1/2 in a geometric progression (GP) with a first term of 64 and a common ratio of 1/2, you need to follow these steps:
Step 1: Determine the formula for the sum of a finite geometric progression. In a GP, the formula for the sum of the first 'n' terms is given by:
Sn = a(1 - r^n) / (1 - r),
where Sn represents the sum of the first 'n' terms, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
Step 2: Substitute the given values into the formula. In this case, a = 64 and r = 1/2. Let the number of terms be 'n'.
Sn = 64(1 - (1/2)^n) / (1 - 1/2).
Step 3: Simplify the equation.
127.5 = 64(1 - (1/2)^n) / (1/2).
Multiply both sides of the equation by (1/2) to get rid of the fraction in the denominator:
127.5 * (1/2) = 64(1 - (1/2)^n).
63.75 = 64(1 - (1/2)^n).
Step 4: Solve for '(1/2)^n' using algebraic manipulation.
Divide both sides of the equation by 64:
63.75 / 64 = 1 - (1/2)^n.
0.995 = 1 - (1/2)^n.
Subtract 1 from both sides of the equation:
0.995 - 1 = - (1/2)^n.
-0.005 = - (1/2)^n.
Take the reciprocal of both sides of the equation:
-200 = (2)^n.
Step 5: Solve for the exponent 'n' by taking the logarithm of both sides of the equation.
log((-200)) = log((2)^n).
Using the logarithm property, ln((-200)) = n * ln(2).
n = ln((-200)) / ln(2).
Step 6: Calculate the value of 'n' using a calculator.
n ≈ -7.643856 / 0.693147.
n ≈ -11.036.
Since the number of terms cannot be negative, we take the positive value of 'n'. Therefore, the number of terms needed to obtain a total of 127 1/2 in the GP is 12 (rounded up from 11.036).