Sum of a series

**Punch** · June 10th 2011, 12:17 AM

Given that $S_n$ is the sum of a geometric series where $S_n=1+\frac{a-2}{4}+\frac{(a-2)^2}{16}+...+\frac{(1-2)^{n-1}}{4^{n-1}}$

Find the values of $a$ such that $256S_4=175S$

$256S_4=175S$

$256(1+\frac{a-2}{4}+\frac{(a-2)^2}{16}+\frac{(a-2)^3}{64})=175+175$

$(1+\frac{a-2}{4}+\frac{(a-2)^2}{16}+\frac{(a-2)^3}{64})=1\frac{47}{128}$

$\frac{a-2}{4}+\frac{(a-2)^2}{16}+\frac{(a-2)^3}{64})=\frac{47}{128}$

$16(a-2)+4(a-2)^2+(a-2)^3=23.5$

**abhishekkgp** · June 10th 2011, 02:37 AM

Originally Posted by Punch

Given that $S_n$ is the sum of a geometric series where $S_n=1+\frac{a-2}{4}+\frac{(a-2)^2}{16}+...+\frac{(1-2)^{n-1}}{4^{n-1}}$

Find the values of $a$ such that $256S_4=175S$

$256S_4=175S$
what is S??

$256(1+\frac{a-2}{4}+\frac{(a-2)^2}{16}+\frac{(a-2)^3}{64})=175+175$
how did you get 175+175 in the RHS??

$(1+\frac{a-2}{4}+\frac{(a-2)^2}{16}+\frac{(a-2)^3}{64})=1\frac{47}{128}$

$\frac{a-2}{4}+\frac{(a-2)^2}{16}+\frac{(a-2)^3}{64})=\frac{47}{128}$

$16(a-2)+4(a-2)^2+(a-2)^3=23.5$

...

**HallsofIvy** · June 10th 2011, 03:39 AM

Originally Posted by Punch

Given that $S_n$ is the sum of a geometric series where $S_n=1+\frac{a-2}{4}+\frac{(a-2)^2}{16}+...+\frac{(1-2)^{n-1}}{4^{n-1}}$

I am going to assume that there should be "a- 2" in that last term.

Find the values of $a$ such that $256S_4=175S$

I am going to assume that "S" is the sum of the infinite series, which is $\frac{4}{6-a}$ .

$256S_4=175S$

$256(1+\frac{a-2}{4}+\frac{(a-2)^2}{16}+\frac{(a-2)^3}{64})=175+175$

How did 175S become 175+ 175? You should have
$256\left(1+ \frac{a-2}{4}+ \frac{(a- 2)^2}{16}+ \frac{a-2)^3}{64}\right)= \frac{700}{6- a}= \frac{700}{4- (a-2)}$
if you let y= a- 2 that becomes
$256\left(1+ \frac{y}{4}+ \frac{y^2}{16}+ \frac{y^3}{64}\right)= \frac{700}{4-y}$
$\left(1+ \frac{y}{4}+ \frac{y^2}{16}+ \frac{y^3}{64}\right)= \frac{175}{64(4- y)}$

Multiplying on both sides by 64:
$64+ 16y+ 4y^2+ y^3= \frac{175}{4-y}$

Multiplying both sides by 4- y:
$256- y^4= 175$
so the whole problem reduces to
$y^4= 81$

$(1+\frac{a-2}{4}+\frac{(a-2)^2}{16}+\frac{(a-2)^3}{64})=1\frac{47}{128}$

$\frac{a-2}{4}+\frac{(a-2)^2}{16}+\frac{(a-2)^3}{64})=\frac{47}{128}$

$16(a-2)+4(a-2)^2+(a-2)^3=23.5$

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