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

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.3 Geometric Sequences; Geometric Series - 11.3 Assess Your Understanding - Page 845: 70

Answer

The sequence is arithmetic with $d=2$. and $S_{50}=2300$

Work Step by Step

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: $S_n=\dfrac{n}{2}[2a_1+(n−1)d] (1)$ where, $a_1 =\ First \ term$, $a_n$ = $n$th term, and $n =\ Number \ of \ Terms$ To identify the sequence as arithmetic or geometric, we substitute $n=1,2,3$ to list the first three terms to obtain: $a_1=(2)(1)-5=-3 \\a_2=(2)(2)-5=-1 \\a_3=(2)(3)-5=1$ We see that the values increase by $1$, so the sequence is arithmetic with $d=a_n-a_{n-1}=a_2-a_1=-1+3=2$. In order to find the sum of the first $50$ terms, we will substitute $a_1=-3$ and $d=2$ in the formula in (1) to obtain: $S_{50}=\dfrac{50}{2} [(2)(-3)+2(50−1)]=25(-6+98)=2300$ Therefore, the sum of first 50 terms is equal to: $S_{50}=2300$
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