Series and Sequences assignment

[grave hazel]

I kinda don't understand

[grand tangle]

What dont you understand

[grave hazel]

the r^100 (asub1 + ....) = 2(1)

[grand tangle]

So we're giving the sum of first 100 terms of a geometric series is 1
We can write it as
a + ar + ar^2 +...+ar^99 = 1
Where a is first term and r is common ratio
we're also given that sun of the next 100 terms i.e the 100 terms after the 100th term is 2
So
ar^100 + ar^101 +...+ar^199 = 2

Now there's two unknowns a and r, two equations
From our second equation we'll take r^100 common , so no it becomes
r^100(a+ar+ar^2+...+ar^99) = 2
Now look at the terms inside the parentheses and compare it with our first equation
You can see that they are the same and their value is just 1
So
r^100 = 2

[grave hazel]

a and r, whats a?

[grand tangle]

A is the first term of the geometric sequence

Unless you mean the value lol

[grave hazel]

ah

What do you mean take r^100 common?

take it as greatest common factor?

[grand tangle]

like if you have 2x + 4y you can write it as 2(x+2y)
If you multiply 2 with (x+ 2y) you get the same answer so we can write it like that

[grave hazel]

ah so basically gcf yeah

But

but why did we not do that for the 1st 100 term

The sum of the second 100 terms is 2, does this basically mean a to ar^199?

[grand tangle]

Because we couldn't take r^something as common as the first term doesn't contain r. (Well we could but it'd be messy). You could take a as common though but the approach would be different.
For the second equation however we can simplify it greatly because if we take r^100 out, we get something in the parentheses whose value we already know

[grand tangle]

no from a^100 to a^199

The sum of the frist 100 terms is 1, the sum of the next 100 terms which means leaving the terms from 100 terms

[grave hazel]

oh ok I see

[grave hazel]

.close