Apopulation of butterflies grow in such a way that each generation is simply 1.5 times the previous generation. there were 350 butterflies in the first generation, how many will there be by the 19th generation?

the formula is an= r* an-1

Hello there!

We know that in the first generation there are 350 butterflies.

then in the second generation, there will be 1.5*350 = 525 butterflies, and so on.

the third generation you have 1.5*(1.5*350) = 788 butterflies

you can see the pattern here, each generation after the first one, you add another 1.5 to the multiplication:

The equation that describes the growth of the population as a function of the generations n, is the next one:

P(n) = 350*1.5^(n -1)

now, the if n = 19, we have:

p(19) = 350*1.5^(19 - 1) = 350*1.5^(18) = 517262.15

but you can't have a 0.15 of a butterfly, you need a whole number, then we need to round it.

The first number after the decimal point is a 1, so we need to round down the population:

The population of butterflies in the 19th generation is equal to 517262 butterflies.

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}

my acc plz

step-by-step explanation: