This is a finite summation of an arithmetic series because each subsequent term is simply subtracting two. We can figure out what the first term of the sequence is. Of course, you could just add up all the terms from 1 to 11, but it is probably better to use or derive a formula. In order to find the sum of an arithmetic sequence, you must find three values:
We know that the first term of a summation is when the variable, n, equals its first possible value, which is one, in this case.
\(a_1=98-2*1=96\) | |
an can be found by using a formula.
\(a_n=a_1+d(n-1)\) | Of course, we already know what a1 equals. We know the common difference is -2. The coefficient of the variable indicates this information. |
\(a_{11}=96-2(11-1)\) | Time to simplify! |
\(a_{11}=96-20\) | |
\(a_{11}=76\) | |
What about the number of terms? Well, this can be determined using the term numbers indicated in summation formula.
\(n=11-1+1=11\)
Let's put all this information together and use the formula to find the sum.
\(S_n=n\left(\frac{a_1+a_n}{2}\right)\)
\(S_n=n\left(\frac{a_1+a_n}{2}\right)\) | Plug in the values and simplify. |
\(S_{11}=11(\frac{96+76}{2})\) | |
\(S_{11}=946\) | |
11A recursive rule allows one to generate future terms of a sequence if one knows one or more of the previous terms. We know that the explicit rule can be written in the following form:
\(a_n=a_1+d(n-1)\)
There are two variables that we must identify in order to finish this formula. Those are:
We can easily identify both of these with some observation. It is given that a1 =21 since that information was given in the recursive formula. In the recursive formula, one must add 9 to obtain the next term in the sequence. This would be the common difference. Let's fill that in and simplify completely.
\(a_n=a_1+d(n-1)\) | Substitute in the known values and simplify. |
\(a_n=21+9(n-1)\) | Distribute the 9 into the binomial. |
\(a_n=21+9n-9\) | Combine like terms. |
\(a_n=9n+12\) | This answer corresponds to the first answer choice. |