Find the minimum number of terms in the geometric series 10 + 8 + 32/5 + ... such that the sum is greater than 49.
Well, if you're throwing around fractions, you must really be into sharing! Let's do some math and share the answer.
The formula to find the sum of a finite geometric series is S = a(1 - r^n) / (1 - r), where 'S' is the sum, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
In this case, the first term 'a' is 10, and the common ratio 'r' is 8/10 = 4/5.
We want the sum to be greater than 49, so we have the inequality:
49 < 10(1 - (4/5)^n) / (1 - 4/5)
Let's simplify that. By multiplying both sides of the inequality by (1 - 4/5), we get:
49(1 - 4/5) < 10(1 - (4/5)^n)
Or, 9 < 10(1 - (4/5)^n)
Dividing both sides by 10, we have:
0.9 < 1 - (4/5)^n
By subtracting 1 and multiplying by -1, we find:
0.1 > (4/5)^n
We can rewrite (4/5)^n as (2/5)^(2n).
So, we have:
0.1 > (2/5)^(2n)
Now, to determine the minimum number of terms, we need to find the smallest value of 'n' that satisfies this inequality. And by "smallest," I mean, the least number of terms needed to make this more complicated than it already is!
To solve this, we need to raise both sides of the inequality to the power of (1/2):
(0.1)^(1/2) > [(2/5)^(2n)]^(1/2)
Taking the square root of 0.1 gives us:
0.316 < (2/5)^n
We can now rewrite (2/5)^n as (0.4)^n.
So, we have:
0.316 < 0.4^n
Now, don't worry, I won't make you calculate this by hand. Here's a little secret: you can use a calculator to find logarithms and prove that the minimum value of 'n' needed to satisfy this inequality is 2.
Therefore, the minimum number of terms needed in the series to make the sum greater than 49 is 2. But hey, don't stop there. Keep challenging those series and see how far you can go!
To find the minimum number of terms in a geometric series such that the sum is greater than 49, we need to find the sum of the series. The formula for the sum of a geometric series is given by:
S = a * (1 - r^n) / (1 - r)
where:
S: sum of the series
a: first term of the series
r: common ratio of the series
n: number of terms
In this case, the first term (a) is 10, and the common ratio (r) is 8/10 = 4/5.
We need to find the smallest value of n such that S is greater than 49.
Using the formula above, we can rewrite it as:
49 < 10 * (1 - (4/5)^n) / (1 - 4/5)
Simplifying it further:
49 < 10 * (1 - (4/5)^n) / (1/5)
Multiplying both sides by 1/5:
49 * 5 < 50 * (1 - (4/5)^n)
245 < 50 - 50 * (4/5)^n
245 - 50 < -50 * (4/5)^n
195 < -50 * (4/5)^n
Now, we need to solve for n. Divide both sides by -50:
195 / -50 > (4/5)^n
-3.9 > (4/5)^n
Taking the logarithm of both sides with base 10:
log(-3.9) > n * log(4/5)
Since the logarithm of a negative number is undefined, there is no solution for n in this case. Therefore, there are no number of terms in the geometric series such that the sum is greater than 49.
To find the minimum number of terms in a geometric series such that the sum is greater than a given value, you can use the formula for the sum of a geometric series:
Sn = a * (r^n - 1) / (r - 1)
where:
Sn is the sum of the first n terms,
a is the first term of the series,
r is the common ratio of the series,
and n is the number of terms.
In this case, we have the geometric series:
10 + 8 + 32/5 + ...
To find the common ratio, r, we can divide any term by the previous term. Let's divide the second term (8) by the first term (10):
r = 8 / 10 = 4/5
Now, we need to find the minimum number of terms, n, such that the sum is greater than 49. We can rewrite the formula for the sum of a geometric series as:
Sn = a * (1 - r^n) / (1 - r)
We want to find the value of n that satisfies the inequality:
Sn > 49
Substituting the values into the inequality, we get:
a * (1 - r^n) / (1 - r) > 49
Plugging in the values, we have:
10 * (1 - (4/5)^n) / (1 - 4/5) > 49
10 * (5/4)^n > 49 * (5/4 - 1)
10 * (5/4)^n > 49/4
Now, we need to solve for n. We can take the logarithm of both sides of the equation to isolate n:
log[(5/4)^n] > log[(49/4) / 10]
n * log(5/4) > log(49/4) - log(10)
n > (log(49/4) - log(10)) / log(5/4)
Using a calculator, we can find that the right-hand side of the equation is -1.651. Since n represents the number of terms, it must be a positive integer. So, we can round up the value of -1.651 to 0. This means that we need at least 1 term for the sum to be greater than 49.
Therefore, the minimum number of terms in the geometric series 10 + 8 + 32/5 + ... such that the sum is greater than 49 is 1.
In this GP, a = 10, r = 4/5
remember that
S(n) = a(1 - r^n)/(1-r)
Just like in your last question, find n so they are equal, then bump it up to the next whole number.