The following series does not go below zero:

2016+2005+1994+...+L
(a) What is the smallest possible value of the last number L ?
(b) Find the sum of the series.

the terms drop by 11 each time

1994 = 181*11 + 3

So, subtracting eleven 181 more times leaves a remainder of 3

using your usual formula,

S184 = 184/2 (2*2016 + 183(-11)) = 185748

Sorry, what is the "usual formula"?

Sn = n/2 (2a+(n-1)d)

Looks like you need to review your AP stuff.

To find the smallest possible value of the last number L in the series, we need to determine the pattern of the series and find the point where it stops going below zero.

Looking at the given series: 2016 + 2005 + 1994 + ...

We can see that each term is obtained by subtracting 11 from the previous term. So, we can write the series as: 2016, 2016 - 11, 2016 - 2 * 11, 2016 - 3 * 11, ...

To find the smallest possible value of L where the series stops going below zero, we need to find the term where it becomes negative.

Let's assume the kth term is the first term that becomes negative. So, we have:

2016 - k * 11 < 0

Now, solve the equation for k:

-k * 11 < -2016

Divide both sides by -11, remembering that dividing by a negative number flips the inequality sign:

k > 2016 / 11

Since k must be an integer, round up the result to the nearest whole number:

k > 183.27 (rounded up to 184)

Therefore, the smallest possible value of L is 2016 - 184 * 11.

To find the sum of the series, we need to determine how many terms are in the series and then use the formula for the sum of an arithmetic series.

To find the number of terms, we can use the formula:

(number of terms) = (last term - first term) / (common difference) + 1

In this case, the first term is 2016, the common difference is -11, and the last term is L (the smallest possible value we found above).

So, the number of terms in the series is:

(number of terms) = (L - first term) / (common difference) + 1 = (L - 2016) / (-11) + 1

Now, substitute the value of L we found above into the equation to calculate the number of terms.

Once you have the number of terms, you can use the formula for the sum of an arithmetic series:

Sum = (number of terms) / 2 * (first term + last term)

Substitute the values of the number of terms, first term, and last term into the formula to calculate the sum.