## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

The sequence is arithmetic with $d=2$. and $S_{50} =2550$
We will check whether the sequence is arithmetic or geometric. (1) A sequence is arithmetic if there exists a common difference $d$ among consecutive terms such that: $d=a_n−a_{n−1}$ The sum of the first n terms of an arithmetic sequence can be computed as: $S_n=\dfrac{n}{2}[2a_1+(n−1)d] (1)$ (2) A sequence is geometric if there exists a common ratio $r$ among consecutive terms such that: $r=\dfrac{a_n}{a_{n−1}}$ The sum of the first n terms of a geometric sequence can be computed as: $S_n= \dfrac{a_1(1-r^n)}{1-r} (2)$ where, $a_1 =\ First \ term$, $a_n$ = $n$th term, and $n =\ Number \ of \ Terms$ Substitute $n=1,2,3$ to list the first three terms: $a_1=3-(\dfrac{2}{3})(1)=\dfrac{7}{3} \\a_2=3-(\dfrac{2}{3})(2)=\dfrac{5}{3} \\a_3=3-(\dfrac{2}{3})(3)=\dfrac{3}{3}=1$ We see that the values increase by $2$, so the sequence is arithmetic with $d=2$. In order to find the sum of the first 50 terms, we will substitute $a_1=2$ and $d=2$ into the formula in (1) to obtain: $S_{50}=\dfrac{50}{2} [(2)(2)+2(50−1)]=25(102)=2550$ Therefore, the sum of first 50 terms is equal to: $S_{50} =2550$