Solving Inflated Series: Sum of First 20 Terms of a=r^k-1

[member38644]

my problem is regarding sequences:

Sum first 20 terms of a=r^k-1

terms are 2, 4/3, 8/9,...
and ratio is 2/3
and r < 1 so its an inflated series
and 2(2/3)^19 = .009021859795

BUT it is an inflated series right? and my answer is the sum of the
first 20 terms which has to be larger than 2 right? so where am I
going wrong?

 

[TD]

a=r^(k-1) is a formula for the k-th term, not for the sum of the first k terms.
The formula for the sum is: $$s_n=t_{1}\frac{1-q^n}{1-q}$$

 

[Architeuthis Dux]

I would first like to clarify that an inflated series refers to a series where the terms increase in value rather than decrease. In this case, the ratio of 2/3 indicates that the terms are decreasing, not inflating. Therefore, the issue here is not with an inflated series, but with the calculation of the sum of the first 20 terms.

To solve this problem, we can use the formula for the sum of a finite geometric series, which 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, we have a = 2, r = 2/3, and n = 20. Plugging these values into the formula, we get:

S = 2(1 - (2/3)^20) / (1 - 2/3)
= 2(1 - 0.00000000000000000000000000000000064) / (1/3)
= 2(0.99999999999999999999999999999999936) / (1/3)
= 0.99999999999999999999999999999999936 * 3
= 2.99999999999999999999999999999999808

As you can see, the sum of the first 20 terms is indeed larger than 2, as expected. Therefore, your answer of 0.009021859795 is incorrect. I would suggest going over the calculation steps again and double-checking your work. It is always a good practice to verify your results to ensure accuracy.

In summary, the issue here is not with an inflated series, but with the calculation of the sum. By using the correct formula and values, we can obtain the correct answer. I hope this explanation helps clarify your doubts and assists you in solving similar problems in the future.