Geometric sequences

Algebra Level 3

If 4 ( 1 − r + r 2 − r 3 + ⋯ ) = 1 + r 2 + r 4 + r 6 + ⋯ 4(1-r+r^{2}-r^{3}+\cdots)=1+r^{2}+r^{4}+r^{6}+\cdots then what is the value of r r to 2 decimal places?


The answer is 0.75.

Freddie Hand by Freddie Hand , 18, United Kingdom

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

4 ( 1 − r + r 2 − r 3 + r 4 − ⋯ ) = 1 + r 2 + r 4 + r 6 + ⋯ 4 ( 1 1 + r ) = 1 1 − r 2 4 ( 1 + r ) = 1 ( 1 − r ) ( 1 + r ) 4 = 1 1 − r 1 − r = 1 4 r = 3 4 = 0.75 \begin{aligned} 4(1-r+r^2 - r^3 + r^4 - \cdots ) & = 1 + r^2 + r^4 + r^6 + \cdots \\ 4\left( \dfrac{1}{1+r} \right) & = \dfrac{1}{1-r^2} \\ \dfrac{4}{ \cancel{(1+r)}} & = \dfrac{1}{ (1-r)\cancel{(1+r)}} \\ 4 & = \dfrac{1}{1-r} \\ 1-r & = \dfrac{1}{4} \\ r & = \dfrac{3}{4} = \boxed{\color{#3D99F6} 0.75} \end{aligned}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

4 ( 1 − r + r 2 − r 3 + . . . ) = 1 + r 2 + r 4 + r 6 + . . . 4 ( ( 1 + r + r 2 + r 3 + . . . ) − 2 ( r + r 3 + r 5 + r 7 + . . . ) ) = 1 + r 2 + r 4 + r 6 + . . . 4 ( ( 1 + r + r 2 + r 3 + . . . ) − 2 r ( 1 + r 2 + r 4 + r 6 + . . . ) ) = 1 + r 2 + r 4 + r 6 + . . . 4 1 − r − 8 r 1 − r 2 = 1 1 − r 2 for ∣ r ∣ < 1 4 ( 1 + r ) 1 − r 2 − 8 r 1 − r 2 = 1 1 − r 2 4 + 4 r − 8 r = 1 ⟹ r = 3 4 = 0.75 \begin{aligned} 4\left(1-r+r^2-r^3+...\right) & = 1+r^2+r^4+r^6+... \\ 4\left((1+r+r^2+r^3+...) - 2(r+r^3+r^5+r^7+...) \right) & = 1+r^2+r^4+r^6+...\\ 4\left((1+r+r^2+r^3+...) - 2r(1+r^2+r^4+r^6+...) \right) & = 1+r^2+r^4+r^6+... \\ \frac 4{1-r} - \frac {8r}{1-r^2} & = \frac 1{1-r^2} & \small \color{#3D99F6} \text{for }|r| < 1 \\ \frac {4(1+r)}{1-r^2} - \frac {8r}{1-r^2} & = \frac 1{1-r^2} \\ 4 + 4r - 8r & = 1 \\ \implies r & = \frac 34 = \boxed{0.75} \end{aligned}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Not for |r|<0 i.e.|r|<1 is correct.please correct it.

Sudhamsh Suraj - 4 years, 3 months ago

Log in to reply

Thanks. Stupid me.

Chew-Seong Cheong - 4 years, 3 months ago

Please try this problem link .

Sudhamsh Suraj - 4 years, 3 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...