Answered: Let, {an} = -5, 10, -20, 40, . . . ,…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.5: The Binomial Theorem

Problem 35E

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Question

Let, {a_n} = -5, 10, -20, 40, . . . , {b_n} = 10, -5, -20, -35, . . . , and {c_n} = -2, 1, - 1/2, 1/4 , . . . . . Find the difference between the sum of the first 10 terms of
{a_n} and the sum of the first 10 terms of {b_n}.

Expert Solution

Step 1

the given sequence are

{a_n} = -5, 10, -20, 40, . . .

{b_n} = 10, -5, -20, -35, . . . ,

we have to find the difference between the sum of the first 10 terms of
{a_n} and the sum of the first 10 terms of {b_n}.

Step 2

$f i r s t l y, w e c h e c k \{a_{n}\} i s a r i t h m e t i c o r g e o m e t r i c . i f \{a_{n}\} h a s a c o m m o n d i f f e r e n c e t h e n t h e d i f f e r e n c e o f c o n s e c u t i v e t e r m s a r e s a m e, t h e n i t i s a n a r i t h m e t i c s e q u e n c e i f \{a_{n}\} h a s a c o m m o m r a t i o t h e n t h e r a t i o o f c o n s e c u t i v e t e r m s a r e s a m e, t h e n i t i s a n g e o m e t r i c s e q u e n c e N o w, f o r \{a_{n}\} f i n d t h e r a t i o o f t h e f i r s t t w o t e r m s a n d t h e n r a t i o o f s e c o n d a n d t h i r d t e r m . \frac{a_{2}}{a_{1}} = \frac{10}{- 5} = - 2 a n d \frac{a_{3}}{a_{2}} = \frac{- 20}{' 10} = - 2 t h e r a t i o b e t w e e n c o n s e c u t i v e t e r m i s s a m e i . e . - 2 s o, \{a_{n}\} i s g e o m e t r i c t h e g e n e r a l t e r m o f g e o m e r t i c s e r i e s n i s a_{n} = a_{1} r^{n - 1} h e r e r = - 2 (c o m m o n r a t i o) a_{1} = - 5 (f i r s t t e r m) \Rightarrow a_{n} = - 5 {(- 2)}^{n - 1} n o w w e f i n d a_{10} p u t n = 10 \Rightarrow a_{10} = - 5 {(- 2)}^{10 - 1} = - 5 {(- 2)}^{9} = 2560$

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