help with a trial question? (1 Viewer)

[Nikkkkkkkk]

hey guys

i'm in the middle of doing a trial paper and this question has me stumped. the solution makes no sense to me so i'd be very grateful if someone is willing to help

sorry i don't know how to put an image into the message so i've attached it as a word doc instead

thanks heaps

 

[Riviet]

1 + 3x + 5x² + 7x³ + 9x⁴ + .... =

1 + (x+2x) + (x²+2x²+2x²+2x²) + (x³+2x³+2x³+2x³) + (x⁴+2x⁴+2x⁴+2x⁴+2x⁴) + ... (using given info)

= 1 + x + x² + x³ + x⁴ + ...

+ 2x + 2x² + 2x³ + 2x⁴ + ...

+ 2x² + 2x³ + 2x⁴ + ...

+ 2x³ + 2x⁴ + ...

etc.

Assuming that |x|<1, then each line forms a series, with the first term bolded in each.

Also note that I've rearranged the terms so that they form many limiting sums.

So we then have

1/(1-x) + 2x/(1-2x + 2x²/(1-x) + 2x³/(1-x) + 2x⁴/(1-x) + ... (using S=a/[1-r] for each limiting sum)

The next line in the solutions just takes a common factor out.

Observe that 2x + 2x² + ... etc. forms another limiting sum with a=2x, r=x.

so a/(1-r) = 2x/(1-x). Add 1 to this since the limiting series starts at 2x, not 1.

The next line is just expanding.

The last line is just getting a common denominator by multiplying the first term, top and bottom by 1-x.

Hope that helps.

 

[Nikkkkkkkk]

wow thanks! you're a genius

i've also noticed that it's 8 days until your birthday... 7 days until mine! lol the fun of an 18th during the hsc...

 

[Riviet]

Pleasure to help. And happy birthday to you!