how many terms in the series 18-16-13-....must be added to give a sum of 44?

I am not certain of the repetition.

starts at 18, down 34, then down 3, ...

I would like to have another term or two. Frankly, if it keeps going down, it cannot be 44, as the highest value will be 18 at the start.

Well, since the first three terms shown add to 47, you got me.

Now, if we continue, with the difference increasing by one each term, we get

18 16 13 10 5 -1 ...
If we add terms 2 through 5 we get 44.

An odd problem, and, I suspect, garbled in transit.

To determine the number of terms in the series 18-16-13-... that must be added to give a sum of 44, we can follow these steps:

1. Observe the pattern: The series appears to be decreasing by a certain amount with each term.

2. Determine the common difference: Subtract each term from the preceding term to find the common difference. In this case, we have:
18 - 16 = 2 (common difference of 2)
16 - 13 = 3 (common difference of 3)

3. Find the nth term formula: In this case, since the common difference is changing, we cannot directly use the arithmetic progression formula. Instead, we need to find a formula for the nth term.

The difference between the terms (common difference) is increasing by 1 each time, so we have a sequence of increasing differences: 2, 3, 4, ...

By applying the formula for the sum of an arithmetic series, we can find the formula for the nth term:
nth term = first term + (n - 1) * common difference

In our case, the first term (18) remains constant, while the common difference changes by 1 with each term.

4. Calculate the number of terms: We need to find the value of n such that the sum of the first n terms equals 44. Set up the equation:

18 + (n - 1) * common difference = 44

5. Rearrange the equation and solve for n:
n * (common difference) = 44 - first term + common difference
n * (common difference) = 44 - 18 + common difference

Simplifying further:
n * (common difference) = 26 + common difference
n * (common difference - 1) = 26

Since we know that the common difference increases by 1 with each term, we can infer that the common difference is greater than 1. Therefore, we can divide both sides of the equation by (common difference - 1) to get the value of n.

6. Solve for n: Divide both sides of the equation by (common difference - 1):
n = 26 / (common difference - 1)

For the given series, the common difference is 2:
n = 26 / (2 - 1) = 26 / 1 = 26

Therefore, you would need to add 26 terms to the series to achieve a sum of 44.