In a geometric progression of first term 5 and common ratio 4/5, what is the
minimum number of terms that must be added for the difference between
the infinite sum and this sum would be less than 0.3?
To find the minimum number of terms that must be added for the difference between the infinite sum of a geometric progression and a given sum to be less than 0.3, we need to calculate the sum of the geometric progression.
The sum of an infinite geometric progression can be computed using the formula:
S = a / (1 - r)
where:
S is the sum of the geometric progression,
a is the first term of the progression,
and r is the common ratio.
In this case, the first term (a) is 5 and the common ratio (r) is 4/5. Plugging these values into the formula, we get:
S = 5 / (1 - 4/5)
S = 5 / (5/5 - 4/5)
S = 5 / (1/5)
S = 5 * 5
S = 25
Now, we need to find the sum of the progression for a certain number of terms, n. The formula for the sum of a geometric progression with n terms is:
Sn = a * (1 - r^n) / (1 - r)
In this case, we want to find the value of n such that the difference between the sum of the infinite geometric progression (25) and the sum of the first n terms is less than 0.3. Mathematically, this can be expressed as:
25 - Sn < 0.3
Substituting the formulas, we have:
25 - (5 * (1 - (4/5)^n) / (1 - 4/5)) < 0.3
Simplifying this equation will allow us to solve for n. However, given the complexity of the equation, it is easier to find the solution through numerical methods or by using a spreadsheet or calculator program.
Using a spreadsheet or calculator, you can evaluate the left side of the equation for different values of n until you find the minimum value of n that satisfies the condition. Start with n = 1 and increment it gradually until the inequality is satisfied.