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Alpha writes the infinite arithmetic sequence10, 8, 6, 4, 2, 0 ,\ldots.Beta writes the infinite geometric sequence9, 6, 4, \frac{8}{3}, \frac{16}{9}, \ldots.Gamma makes a sequence whose n^{\text{th}} term is the product of the n^{\text{th}} term of Alpha's sequence and the n^{\text{th}} term of Beta's sequence:10\cdot 9 \quad,\quad 8\cdot 6\quad ,\quad 6\cdot 4\quad,\quad 4\cdot \frac83\quad,\quad 2\cdot \frac{16}{9}\quad,\quad \ldo...What is the sum of Gamma's entire sequence?

 Jan 25, 2015

Best Answer 

 #4
avatar+27557 
+10

How about the following:

 

summations

 

Perhaps I should clarify a step:  Note that n*rn = r*n*rn-1= r*d(rn)/dr  (or perhaps this just muddies the waters - in which case ignore it!)

.

.

 Jan 26, 2015
 #1
avatar+99352 
+5

10,8,6,       AP    a=10   d=-2      Tn=a+(n-1)d

 

$$T_{n+1}=a+nd=10-2n$$   (I have changed n to start at 0)

 

9,6,4,......    GP   a=9     r=2/3       Tn=ar^(n-1)

 

$$\\T_{n+1} = ar^n\\\\
T_{n+1}=9*(\frac{2}{3})^n$$

 

Sum of Gamma's Sequence

 

$$\\S_\infty =\displaystyle\sum_0^\infty\quad (10-2n)*9*\frac{2^n}{3^n}\\\\
S_\infty =\displaystyle\sum_0^\infty\quad 90*\frac{2^n}{3^n}\quad-\quad\displaystyle\sum_0^\infty\quad (18n)*\frac{2^n}{3^n}\\\\
S_\infty =90*\displaystyle\sum_0^\infty\quad \frac{2^n}{3^n}\quad-\quad18\displaystyle\sum_0^\infty\quad \frac{n*2^n}{3^n}\\\\$$

 

     $$\\NOW\\\\
\displaystyle\sum_0^\infty\;\;\frac{2^n}{3^n}
=\displaystyle\sum_0^\infty\;\;\left(\frac{2}{3}\right)^n\\\\
\qquad $ This is a GP a=1 r=2/3$\\\\
=\frac{a}{1-r}=\frac{1}{1-\frac{2}{3}}=\frac{1}{\frac{1}{3}}=3$$

 

$$\\S_\infty =90*\displaystyle\sum_0^\infty\quad \frac{2^n}{3^n}\quad-\quad18\displaystyle\sum_0^\infty\quad \frac{n*2^n}{3^n}\\\\
S_\infty =90*3\;\;-\quad18\displaystyle\sum_0^\infty\quad \frac{n*2^n}{3^n}\\\\
S_\infty =270\;\;-\quad18\displaystyle\sum_0^\infty\quad \frac{n*2^n}{3^n}\\\\$$

 

Here I cheated a little and used Wolfram|Alpha to solve the second sum.  The answer is 6

The hyperlink is below    

Wolfram|Alpha Second sum solution

 

$$\\S_\infty =270\;\;-\quad18\displaystyle\sum_0^\infty\quad \frac{n*2^n}{3^n}\\\\
S_\infty =270-18*6\\\\
S_\infty = 270-108\\\\
S_\infty = 162\\\\$$

.
 Jan 26, 2015
 #2
avatar+99352 
0

If another mathematician can explain to me how to do the second sum solution that would be good.

 

Ie      Why does

 $$\displaystyle\sum_{n\rightarrow0}^\infty\quad \frac{n*2^n}{3^n}=6\qquad ?$$

 

Thank you     

 Jan 26, 2015
 #3
avatar+21860 
+10

Alpha writes the infinite arithmetic sequence10, 8, 6, 4, 2, 0 ,\ldots.

Beta writes the infinite geometric sequence9, 6, 4, \frac{8}{3}, \frac{16}{9}, \ldots.

Gamma makes a sequence whose n^{\text{th}} term is the product of the n^{\text{th}} term of Alpha's sequence and the n^{\text{th}} term of Beta's sequence:10\cdot 9 \quad,\quad 8\cdot 6\quad ,\quad 6\cdot 4\quad,\quad 4\cdot \frac83\quad,\quad 2\cdot \frac{16}{9}\quad,\quad \ldo...

What is the sum of Gamma's entire sequence ?

 

$$\\\small{\text{
Sequence alpha:
$ a_n = a_1 + (n-1)*d = (a_1-d) + n*d
\quad a_1=10$ and $d=8-10=a_{n+1}-a_n=-2$
}}\\
\small{\text{
Sequence beta:
$ b_n = b_1 *r^{n-1} \quad b_1=9$ and $r=\frac{6}{9}=\frac{b_{n+1}}{b_n}=\frac{2}{3}$
}}$\\\\$
\small{\text{
Sequence gamma :
$ g_n = a_n*b_n = b_1 *r^{n-1} [(a_1-d) + n*d]
$
}}\\
\small{\text{
$ \boxed{ g_n = \underbrace{ \left[ b_1(a_1-d) \right] *r^{n-1} }_{sum\ = \dfrac{b_1(a_1-d)}{1-r} } \ +\ b_1d* n r^{n-1} }
$ The sequence of Gamma has two parts.
}}$\\\\$
\small{\text{
The sum of the first part $ \left[ b_1(a_1-d) \right] *r^{n-1} $ is the sum of a geometric sequence $
= \frac{ b_1(a_1-d)}{1-r}
$
}}$\\\\$
\small{\text{
The sum s of the second part $ b_1d* n r^{n-1} $ is:
}} \\
\begin{array}{rcrrrrr}
s & = & (b_1d) * 1 * r^0 +&
(b_1d)* 2 * r^1 \ + &(b_1d) * 3 * r^2 \ +&(b_1d) * 4 * r^3 \ + &(b_1d) * 5 * r^4 \ + \dots \\
r*s & = & & (b_1d)* 1 * r^1 \ + &(b_1d) * 2 * r^2 \ +&(b_1d) * 3 * r^3 \ + &(b_1d) * 4 * r^4 \ + \dots \\
\hline
s-r*s & = & (b_1*d) \ + & (b_1d)*r^1 \ + &(b_1d)*r^2 \ +&(b_1d)*r^3 \ +&(b_1d)*r^4 \ + \dots \\
\end{array}\\
\small{\text{
$
s-r*s = \underbrace{ (b_1*d) \ + (b_1d)*r^1 \ + (b_1d)*r^2 \ +(b_1d)*r^3 \ +(b_1d)*r^4 \ + \dots }_{\text{sum of a geometric sequence }\ = \frac{b_1d}{1-r} }
$
} $\\$
\small{\text{
$
s-r*s = \frac{b_1d}{1-r}
$
}}$\\$
\small{\text{
$
s(1-r) = \frac{b_1d}{1-r}
$
}}$\\\\$
\small{\text{
$
s = \dfrac{b_1d}{(1-r)^2}
$
}}$\\\\$
\small{\text{
The sum of Gamma's sequence
$ = \dfrac{b_1(a_1-d)}{1-r} \ + \dfrac{b_1d}{(1-r)^2}
$
}}$\\\\$
\small{\text{
$ = \left( \dfrac{b_1}{1-r} \right) * \left[ (a_1-d)+\dfrac{d}{(1-r)} \right]
$
}}$$

$$\small{\text{
The sum of Gamma's sequence
$ = \left(
\dfrac{ 9 }{ 1 - \frac{2}{3} }
\right) *
\left[ (10-(-2))+ \dfrac{ (-2) } { 1 - \frac{2}{3} }
\right] $
}}\\
\small{\text{
$ = 9*3 *(12-2*3) = 27*6 = 162
$
}}$$

.
 Jan 26, 2015
 #4
avatar+27557 
+10
Best Answer

How about the following:

 

summations

 

Perhaps I should clarify a step:  Note that n*rn = r*n*rn-1= r*d(rn)/dr  (or perhaps this just muddies the waters - in which case ignore it!)

.

.

Alan Jan 26, 2015
 #5
avatar+99352 
0

Thanks Guys! 

 Jan 27, 2015
 #6
avatar+98184 
0

Heureka....I'm trying to learn more about how to manipulate these series questions....I followed everything except for this.....

How did you get from [b1(a1 - d)]*rn-1    to     [b1(a1 - d)] / (1 - r)    ???

Thanks in advance.....

 

 Jan 27, 2015
 #7
avatar+21860 
+5

How did you get from [b1(a1 - d)]*rn-1    to     [b1(a1 - d)] / (1 - r)    ???

Hi CPhill,

first i tell you about der geometric sequence of beta:

$$\small{\text{
The geometric sequence is
$
b_1 = \underbrace{9*(\frac{2}{3})^0}_{=9}, \ b_2 = \underbrace{9*(\frac{2}{3})^1}_{=6}, \ b_3 = \underbrace{9*(\frac{2}{3})^2}_{=4}, \ b_4 = \underbrace{9*(\frac{2}{3})^3}_{=\frac{8}{3}}, \ \dots \ ,\ b_n = 9*(\frac{2}{3})^{n-1}
$
}} \\
\small{\text{
The sum $s=\frac{9}{1-\frac{2}{3}}$\ , now we set $\ b_1 = 9 $ and $\ r = \frac{2}{3}
$ and see why
}}\\\\
\small{\text{
b_1 = b_1, \ b_2 = b_1 r, \ b_3 = b_1r^2, \ b_4 = b_1r^3, \ \dots \ ,\ b_n = b_1r^{n-1}
$
}} \\
\small{\text{
The sum is
$
s = b_1 + b_1 r + b_1r^2 + b_1*r^3 + b_1r^4 + \ \dots
$
}} \\
\small{\text{
$
\begin{array}{rcll}
\hline
s &=& b_1 \ + & b_1 r + b_1r^2 + b_1*r^3 + b_1r^4 + \ \dots \\
rs &=& & b_1 r + b_1r^2 + b_1*r^3 + b_1r^4 + \ \dots \\
\hline
s-rs &=& b_1 & + 0 + 0 + 0 + 0 +\dots \\
s(1-r) &=& b_1 & \\
s &=& \dfrac{b_1}{1-r} &
\end{array}
$
}} \\
\boxed{
\small{\text{
So, if we have a Geometric Sequence $ b_n = b_1r^{n-1} $ the infinite sum $ s = \dfrac{b_1}{1-r} $
}}
}$$

 

Now we look to Gamma Sequence.  There is a constant:    $$b_1(a_1-d)=g_1$$

  $$\boxed{
\small{\text{
The Geometric Sequence Gamma first part is $ g_{n_{First Part}} = g_1r^{n-1} $ the infinite sum $ s = \dfrac{g_1}{1-r} $
}}
}\\\\
\small{\text{
$ s = \dfrac{g_1}{1-r} =\dfrac{b_1(a_1-d)}{1-r}$
}}\\\\
\small{\text{
set $ b_1 = 9, \ a_1 = 10,\ d = -2, \ r =\frac{2}{3}
$
}}\\
\small{\text{
we have the sum of Gamma Sequence first part:
}}\\
\small{\text{
$
108*(\frac{2}{3})^0 + 108*(\frac{2}{3})^1 + 108*(\frac{2}{3})^2 +108*(\frac{2}{3})^3 + \dots
$
}} \\
\small{\text{
and the infinite sum $s = \frac{108}{1-\frac{2}{3}}$
and $g_{1_{first part}}=108,\ g_{2_{first part}}=108*\frac{2}{3}, \ \dots \ ,\ g_{n_{first part}}=108*(\frac{2}{3})^{n-1} $
}}\\$$

P.S.

$$\\\small{\text{
The finite sum:
$
s_n=a_1 + a_1\cdot r + a_1\cdot r^2 + a_1\cdot r^3 + \dots +a_1\cdot r^{n-1}
$
is
$
s_n = \dfrac{a_1(1-r^n)}{1-r} $
}}\\\\
\small{\text{
The infinite sum:
$
s_n=a_1 + a_1\cdot r + a_1\cdot r^2 + a_1\cdot r^3 +a_1\cdot r^4 \dots
$
}}\\\\
\small{\text{
with
$
s_n = \dfrac{a_1(1-r^n)}{1-r} $
is $s=\lim\limits_{n\to \infty} s_n = \dfrac{a_1}{1-r}$,}}\\\\
\small{\text{
if $\quad -1 }}$$

.
 Jan 27, 2015
 #8
avatar+99352 
0

Thanks Heureka  

 Jan 28, 2015

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