Answered: 1024 2*5 25 + + 48828125

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter4: Polynomials

Section4.4: Powers Of Monimials

Problem 7MRE

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Question

Given a partial sum of a geometric sequence:
a. Find a formula for the nth term, ??, explicitly.
b. Find the exact value for the partial sum.

Expert Solution

Step 1

The general form of Geometric Progression is:

a, ar, ar², ar³, ar⁴,…,aⁿ

Where,

a = First term

r = common ratio

The common ratio is defined as when we divide the term by its previous term then the value obtained is termed as a common ratio.

aⁿ = nth term

The formula for the nth term of the g.p is given by

^{$a_{n} = a {(r)}^{n - 1}$ .......(1)}

Step 2

Here, the given sequence is

$\frac{1}{2} + \frac{1}{5} + \frac{2}{25} + . . . . . + \frac{1024}{48828125}$

The first term is $a = \frac{1}{2}$ .

The common ratio is $r = \frac{\frac{1}{5}}{\frac{1}{2}} = \frac{2}{5}$ .

The formula for the nth term of the g.p is given by

^{$a_{n} = a {(r)}^{n - 1} \frac{1024}{48828125} = \frac{1}{2} {(\frac{2}{5})}^{n - 1} \frac{2048}{48828125} = {(\frac{2}{5})}^{n - 1} {(\frac{2}{5})}^{11} = {(\frac{2}{5})}^{n - 1}$}

^{Equating the power of the above equation,}

^{$n - 1 = 11 n = 12$}

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