Q. 10 Determine whether the given sequ... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 12: Q. 10 (page 839)

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms.
$25, 10, 4, \frac{8}{5}, . . .$

Short Answer

Expert verified

$25, 10, 4, \frac{8}{5}, . . .$ is a Geometric Sequence.

Also, Common Ratio, $r = \frac{2}{5}$ and Sum of first n terms, $S_{n} = \frac{125}{3} [1 - {(\frac{2}{5})}^{n}]$ .

Step by step solution

Given information:

Consider the given sequence, $25, 10, 4, \frac{8}{5}, . . .$

Formulas:

$F o r A r i t h m e t i c S e q u e n c e : C o m m o n d i f f e r e n c e : d = a_{n} - a_{n - 1} F o r G e o m e t r i c S e q u e n c e : C o m m o n r a t i o : r = \frac{a_{n}}{a_{n - 1}} S_{n} = a_{1} (\frac{1 - r^{n}}{1 - r})$

To check whether the given sequence is arithmetic sequence:

To check for arithmetic: Difference of any 2 successive terms is constant, that is common difference $d .$

The given sequence $25, 10, 4, \frac{8}{5}, . . .$

$d = 10 - 25 = - 15 d = 4 - 10 = - 6$

$鈭�$ The given sequence is not an arithmetic sequence.

To check whether the given sequence is geometric sequence:

To check for geometric: Ratio of successive terms is constant, that is common ratio $r .$

The given sequence is $25, 10, 4, \frac{8}{5}, . . .$

$r = \frac{10}{25} = \frac{2}{5} r = \frac{4}{10} = \frac{2}{5} r = \frac{\frac{8}{5}}{4} = \frac{8}{5} 脳 \frac{1}{4} = \frac{2}{5}$

$鈭�$ The given sequence is a geometric sequence.

And $C o m m o n R a t i o, r = \frac{2}{5} .$

to find the sum of first n terms Sn:

The given sequence is $25, 10, 4, \frac{8}{5}, . . .$

$a_{1} = 25, r = \frac{2}{5}$

role="math" localid="1647508244083" $S_{n} = 25 [\frac{1 - {(\frac{2}{5})}^{n}}{1 - \frac{2}{5}}] = 25 [\frac{1 - {(\frac{2}{5})}^{n}}{\frac{(1 脳 5) - 2}{5}}] = 25 [(1 - {(\frac{2}{5})}^{n}) 脳 \frac{5}{3}] = \frac{125}{3} [1 - {(\frac{2}{5})}^{n}]$

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91影视

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms.
$25, 10, 4, \frac{8}{5}, . . .$

Short Answer

Step by step solution

Given information:

Formulas:

To check whether the given sequence is arithmetic sequence:

To check whether the given sequence is geometric sequence:

to find the sum of first n terms Sn:

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91影视

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms.25,10,4,85,...

Short Answer

Step by step solution

Given information:

Formulas:

To check whether the given sequence is arithmetic sequence:

To check whether the given sequence is geometric sequence:

to find the sum of first n terms Sn:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Geometry

Decision Maths

Logic and Functions

Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms.
$25, 10, 4, \frac{8}{5}, . . .$