Question

The total number of fungal spores can be found using an infinite geometric series where a1 = 10 and the common ratio is 3. Find the sum of this infinite series that will be the upper limit of the fungal spores.

495

990

1,485

The series is divergent

Answer 1

the series doesn't have an infinite sum as the common ratio is 3.

for a series to be convergent then r needs to met the following condition

\[-1 < r < 1\]

hope it helps

Answer 2

ok but how would i use that in this equation.

Answer 3

@campbell_st typo? *does have*, or *finite"

Answer 4

i still dont see how i use that for this equation

Answer 5

The common ratio is 3, which is greater than 1.

The spore population is given by a series,
\[\large\sum_{k=0}^na3^k\]
When \(k=0\), you have the initial amount/starting number of spores, \(a3^0=a\).

When \(k=1\), you have \(a3^1=3a\). The sum of these first two terms is \(4a\).

When \(k=2\), you have \(a3^2=9a\). The sum is \(4a+9a=13a\).

And so on. You can keep plugging in successive values of \(k\) and add up the terms, you'll find that the spore count just keeps going up and up. This means there's no upper limit to the spore count.

This means the series diverges. It's a direct consequence of the fact that the common ratio is not between -1 and 1, as @campbell_st said.