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1-16+246-4096+...+4294967296

Guest Oct 28, 2015

Best Answer 

 #1
avatar+20680 
+30

1-16+246-4096+...+4294967296

 

 

If you mean:  1 - 16 + 256 - 4096 +-... + 4294967296          256 instead of 246 !!!

We have:

\(\begin{array}{rcl} a_1&=&1\\ a_2&=&- 16 \\ a_3&=&256 \\ a_4&=&- 4096 \\ \dots \\ a_9 &=& 4294967296\\ \boxed{~ r = \frac{a_{n}}{a_{n-1}} ~}\\ r&=&\frac{ a_4 }{ a_3 }=\frac{ a_3 }{ a_2 }=\frac{ a_2 }{ a_1 }\\ r&=&\frac{ - 4096}{ 256 }=\frac{ 256 }{ - 16 }=\frac{ - 16 }{ 1 }\\ r&=& - 16= - 16= - 16\\ \mathbf{r} &\mathbf{=}& \mathbf{- 16}\\ \boxed{~ a_n = a_1\cdot r^{n-1} \\ a_n = a_1\cdot (-16)^{n-1} ~}\\ a_1 &=& 1\cdot \left( - 16 \right)^{1-1} = 1\\ a_2 &=& 1\cdot \left( - 16 \right)^{2-1} = 1\cdot - 16= - 16\\ a_3 &=& 1\cdot \left( - 16 \right)^{3-1} = 1\cdot \left( - 16 \right)^2 = 256\\ a_4 &=& 1\cdot \left( - 16 \right)^{4-1} = 1\cdot \left( - 16 \right)^3 = - 4096 \\ \dots \\ a_{9} &=& 1\cdot \left(- 16 \right)^{9-1} = 1\cdot \left( - 16 \right)^8 = 4294967296\\ \boxed{~ sum_n = a_1\cdot\frac{1-r^n}{1-r} ~}\\ sum_9 &=& 1 \cdot \frac{1-(-16)^9}{1-(-16)} \\ sum_9 &=&\frac{1+68\ 719\ 476\ 736}{17} \\ sum_9 &=&\frac{68\ 719\ 476\ 737}{17} \\ \mathbf{sum_9} &\mathbf{=}& \mathbf{4\ 042\ 322\ 161}\\ \end{array}\)

 

1 - 16 + 256 - 4096 +-... + 4294967296 = 4 042 322 161

 

laugh

heureka  Oct 29, 2015
edited by heureka  Oct 29, 2015
 #1
avatar+20680 
+30
Best Answer

1-16+246-4096+...+4294967296

 

 

If you mean:  1 - 16 + 256 - 4096 +-... + 4294967296          256 instead of 246 !!!

We have:

\(\begin{array}{rcl} a_1&=&1\\ a_2&=&- 16 \\ a_3&=&256 \\ a_4&=&- 4096 \\ \dots \\ a_9 &=& 4294967296\\ \boxed{~ r = \frac{a_{n}}{a_{n-1}} ~}\\ r&=&\frac{ a_4 }{ a_3 }=\frac{ a_3 }{ a_2 }=\frac{ a_2 }{ a_1 }\\ r&=&\frac{ - 4096}{ 256 }=\frac{ 256 }{ - 16 }=\frac{ - 16 }{ 1 }\\ r&=& - 16= - 16= - 16\\ \mathbf{r} &\mathbf{=}& \mathbf{- 16}\\ \boxed{~ a_n = a_1\cdot r^{n-1} \\ a_n = a_1\cdot (-16)^{n-1} ~}\\ a_1 &=& 1\cdot \left( - 16 \right)^{1-1} = 1\\ a_2 &=& 1\cdot \left( - 16 \right)^{2-1} = 1\cdot - 16= - 16\\ a_3 &=& 1\cdot \left( - 16 \right)^{3-1} = 1\cdot \left( - 16 \right)^2 = 256\\ a_4 &=& 1\cdot \left( - 16 \right)^{4-1} = 1\cdot \left( - 16 \right)^3 = - 4096 \\ \dots \\ a_{9} &=& 1\cdot \left(- 16 \right)^{9-1} = 1\cdot \left( - 16 \right)^8 = 4294967296\\ \boxed{~ sum_n = a_1\cdot\frac{1-r^n}{1-r} ~}\\ sum_9 &=& 1 \cdot \frac{1-(-16)^9}{1-(-16)} \\ sum_9 &=&\frac{1+68\ 719\ 476\ 736}{17} \\ sum_9 &=&\frac{68\ 719\ 476\ 737}{17} \\ \mathbf{sum_9} &\mathbf{=}& \mathbf{4\ 042\ 322\ 161}\\ \end{array}\)

 

1 - 16 + 256 - 4096 +-... + 4294967296 = 4 042 322 161

 

laugh

heureka  Oct 29, 2015
edited by heureka  Oct 29, 2015
 #2
avatar+94114 
0

That is some good forensic maths there Heureka     wink laugh cool

Melody  Oct 29, 2015

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