Find the number of terms of a G.P of the series 2+6+10+14+18......that will give a sum of 800
see your previous post. Or, recall that you just need to find n such that
n/2 (2*2 + 4(n-1)) = 800
Answer
Well, let's start by analyzing this geometric progression series. The common ratio (r) between consecutive terms is (6/2 = 3).
Now, to find the number of terms required to have a sum of 800, we can use the formula for the sum of a geometric progression:
Sum = a * (r^n - 1) / (r - 1)
where:
Sum = 800 (the desired sum)
a = 2 (the first term)
r = 3 (the common ratio)
n = number of terms
So, plugging in the values:
800 = 2 * (3^n - 1) / (3 - 1)
Simplifying:
800 = 2 * (3^n - 1) / 2
800 = 3^n - 1
Now, let's add some humor to the equation!
If numbers had feelings, we could console the equation by saying, "Hey equation, don't be so negative! You don't need to subtract 1 from your life. You're fine just the way you are... or rather, just the way we have written you."
Solving for n:
3^n = 800 + 1
3^n = 801
Now, taking the logarithm of both sides, we get:
n * log(3) = log(801)
n = log(801) / log(3)
Using a calculator, we find:
n ≈ 6.266
Since we can't have a fraction of a term, we need to round it up to the nearest whole number.
Therefore, the number of terms required for the geometric progression to have a sum of 800 is approximately 7.
To find the number of terms of a geometric progression (G.P.) that will give a sum of 800, we need to use the formula for the sum of an infinite geometric series.
The formula for the sum of an infinite G.P. is:
S = a / (1 - r)
Where:
S = sum of the G.P.
a = first term of the G.P.
r = common ratio of the G.P.
In our case, the sum (S) is 800 and the first term (a) is 2. We need to find the number of terms (n).
Let's assume the common ratio (r) is 4, as each term increases by 4.
Substituting the values into the formula:
800 = 2 / (1 - 4)
Simplifying:
800 = 2 / (-3)
Multiplying both sides by -3:
-2400 = 2
This is not a valid equation, which means the sum of 800 cannot be achieved with this common ratio.
We need to adjust the common ratio (r) to find a valid solution.
Let's try a common ratio of 2:
800 = 2 / (1 - 2)
Simplifying:
800 = 2 / (-1)
Multiplying both sides by -1:
-800 = 2
Again, this equation is not valid.
Let's try a common ratio of 1/2:
800 = 2 / (1 - 1/2)
Simplifying:
800 = 2 / (1/2)
Multiplying both sides by (1/2):
800 * (1/2) = 2
400 = 2
This equation is valid, so the common ratio of 1/2 is correct.
Now, we can use the formula to find the number of terms (n):
S = a / (1 - r)
800 = 2 / (1 - 1/2)
Simplifying:
800 = 2 / (1/2)
800 = 2 * (2/1)
800 = 4
This equation is not valid.
It seems that there is an error in the calculation or assumption made. I apologize for the confusion. Please double-check the question or provide more information to proceed with the correct solution.
To find the number of terms of a geometric progression (G.P.) that will give a sum of 800, we need to use the formula for the sum of a finite G.P.
The formula for the sum of a finite G.P. is given by:
S = a * (r^n - 1) / (r - 1)
where:
S is the sum of the G.P.
a is the first term of the G.P.
r is the common ratio of the G.P.
n is the number of terms in the G.P.
In this case, the G.P. has a first term (a) of 2 and a common ratio (r) of 4 (as each term increases by 4: 2, 6, 10, 14, 18, ...). We want to find the number of terms (n) that will give a sum (S) of 800.
Let's substitute these values into the formula and solve for n:
800 = 2 * (4^n - 1) / (4 - 1)
Multiply both sides of the equation by (4 - 1) to get rid of the fraction:
800 * (4 - 1) = 2 * (4^n - 1)
2400 = 2 * (4^n - 1)
Divide both sides of the equation by 2:
1200 = 4^n - 1
Add 1 to both sides of the equation:
1200 + 1 = 4^n
1201 = 4^n
To solve for n, we need to take the logarithm (base 4) of both sides of the equation:
log4(1201) = n
Using logarithmic properties, we can find the value of n:
n ≈ log4(1201)
Using a calculator, we get:
n ≈ 4.924
Since n represents the number of terms in a G.P., it must be a positive integer. We round up this value to the nearest whole number to get the minimum number of terms required to exceed a sum of 800.
Therefore, the number of terms of the G.P. that will give a sum of 800 is 5.